diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences.py index 81abca2..075aafc 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences.py +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences.py @@ -50,6 +50,7 @@ from kempe_up_tooth_sequences import ( _positions, canonical_sequence, compact_coloring, + dihedral_reading_sequences, seq_str, ) @@ -101,9 +102,10 @@ class Config: class Experiment: - def __init__(self, n: int, m: int): + def __init__(self, n: int, m: int, dihedral: bool = True): self.n = n self.m = m + self.dihedral = dihedral # read sequences off the un-deduped census self.configs: list[Config] = [] for g in generate(n, min_up_teeth=3, dedup=True): for face, edges in inner_face_singletons(g).items(): @@ -118,15 +120,20 @@ class Experiment: self.by_sequence: dict[tuple[int, ...], list[tuple[int, Coloring]]] = defaultdict(list) for cidx, cfg in enumerate(self.configs): - entries: list[tuple[Coloring, tuple[int, ...]]] = [] + entries: list[tuple[Coloring, frozenset]] = [] seqs: set[tuple[int, ...]] = set() for coloring, verdict in classify_colorings(cfg.graph, dedup_colors=True): if not verdict.valid: continue - cseq = canonical_sequence(cfg.sequence(coloring)) - entries.append((coloring, cseq)) - seqs.add(cseq) - self.by_sequence[cseq].append((cidx, coloring)) + if self.dihedral: + cseqs = dihedral_reading_sequences(cfg.graph.n, coloring, + cfg.edges, "d") + else: + cseqs = {canonical_sequence(cfg.sequence(coloring))} + entries.append((coloring, frozenset(cseqs))) + for cseq in cseqs: + seqs.add(cseq) + self.by_sequence[cseq].append((cidx, coloring)) self.colorings.append(entries) self.config_seq_sets.append(frozenset(seqs)) @@ -223,17 +230,19 @@ def write_sequence_note(exp: Experiment, seq, path, fig_name): lines.append("") lines.append( f"Canonical colour sequence of the singleton down-tooth apexes on a " - f"single inner non-tooth face (read in increasing annular-edge order, " - f"reduced modulo the six colour permutations) for Kempe-balanced " - f"3-colourings of M(T) with **n = {exp.n}**, **m = {exp.m} singleton " - f"down apexes on the face**." + f"single inner non-tooth face (read in cyclic order, reduced modulo the " + f"six colour permutations) for Kempe-balanced 3-colourings of M(T) with " + f"**n = {exp.n}**, **m = {exp.m} singleton down apexes on the face**. " + f"Sequences are read off the un-deduped census (every cyclic orientation " + f"of each colouring), so one colouring may realise several sequences -- " + f"see `../kempe_sequence_orientation_note.md`." ) lines.append("") lines.append(f"- Colour multiset: {counts}.") lines.append(f"- Realised by **{len(by_config)}** of {len(exp.configs)} " f"configs (M(T), inner face).") lines.append(f"- **{len(exp.by_sequence[seq])}** Kempe-balanced colourings " - f"(mod colour permutation) produce it.") + f"(mod colour permutation) realise it in some orientation.") lines.append(f"- Figure: `{fig_name}` (black rings mark the face's down apexes).") lines.append("") lines.append("Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` " @@ -250,7 +259,8 @@ def write_sequence_note(exp: Experiment, seq, path, fig_name): lines.append("") for coloring in cols: raw = seq_str(cfg.sequence(coloring)) - lines.append(f"- face apexes (raw labels) `{raw}` → canonical `{s}` · " + lines.append(f"- face apex colours (canonical-rep edge order) `{raw}`, " + f"realises `{s}` in some orientation · " f"`{compact_coloring(cfg.graph, coloring)}`") lines.append("") with open(path, "w") as fh: @@ -263,15 +273,19 @@ def write_summary(exp: Experiment, path): lines.append(f"# Inner-face singleton-down-apex sequences of Kempe-balanced " f"colourings (n={exp.n}, m={exp.m})") lines.append("") + reading = ("the un-deduped census (every cyclic orientation of each " + "colouring)" if exp.dihedral + else "a single anchored representative per dihedral class") lines.append( f"Every full medial tire graph M(T) with |A(T)| = {exp.n} (one " f"representative per dihedral class) that has an inner non-tooth face " f"holding exactly {exp.m} singleton down-tooth apexes: " f"**{len(exp.configs)} configs (M(T), inner face)**. For each we " f"enumerate the Kempe-balanced (valid) proper 3-colourings (modulo " - f"colour permutation), read the down-apex colour sequence in increasing " - f"annular-edge order, and reduce it modulo colour permutation (NOT " - f"dihedral symmetry)." + f"colour permutation), read the down-apex colour sequence in cyclic " + f"order off {reading}, and reduce it modulo colour permutation (NOT " + f"dihedral symmetry). Reading off the census makes the recorded " + f"vocabulary orientation-honest; see `../kempe_sequence_orientation_note.md`." ) lines.append("") total = sum(len(c) for c in exp.colorings) @@ -349,7 +363,9 @@ def write_summary(exp: Experiment, path): # --------------------------------------------------------------------------- def run(args): - exp = Experiment(args.n, args.m) + exp = Experiment(args.n, args.m, dihedral=not args.anchored) + mode = "anchored" if args.anchored else "census (orientation-honest)" + print(f"reading: {mode}") print(f"n={args.n}, m={args.m}: {len(exp.configs)} configs (M(T), inner face)") print(f"distinct canonical down-apex sequences: {len(exp.by_sequence)}") for seq in exp.sequences(): @@ -377,6 +393,9 @@ def main(): parser.add_argument("--m", type=int, default=3, help="singleton down apexes on the inner face (>=3)") parser.add_argument("--no-figures", action="store_true") + parser.add_argument("--anchored", action="store_true", + help="read one anchored representative per class " + "(old behaviour) instead of the un-deduped census") run(parser.parse_args()) diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.md index d019b39..1901d74 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.md @@ -1,10 +1,10 @@ # Inner-face down-apex sequence `012` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 3 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 3 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 1×colour0, 1×colour1, 1×colour2. - Realised by **26** of 26 configs (M(T), inner face). -- **241** Kempe-balanced colourings (mod colour permutation) produce it. +- **241** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_012.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. @@ -13,370 +13,370 @@ Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D 22 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u5:1] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u4:0 u5:1] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u4:0 u5:2] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2 u4:2 u5:2] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]` ## C01 — word=UUUUUDUDD bites=- face=root apexes=[d5,d7,d8] 11 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u6:2] D[d5:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u6:1] D[d5:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u6:1] D[d5:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u4:0 u6:1] D[d5:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u4:0 u6:2] D[d5:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u6:2] D[d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u6:1] D[d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u4:0 u6:1] D[d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u4:0 u6:1] D[d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u4:0 u6:2] D[d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u4:2 u6:2] D[d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]` ## C02 — word=UUUUDUUDD bites=- face=root apexes=[d4,d7,d8] 17 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u5:2 u6:2] D[d4:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u5:1 u6:1] D[d4:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u6:0] D[d4:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u5:1 u6:0] D[d4:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u5:2 u6:0] D[d4:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u5:2 u6:2] D[d4:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u5:1 u6:1] D[d4:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u6:0] D[d4:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u5:1 u6:0] D[d4:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u5:2 u6:0] D[d4:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2 u5:2 u6:2] D[d4:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]` ## C03 — word=UUUUDUDUD bites=- face=root apexes=[d4,d6,d8] 26 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:2 d6:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:1 d6:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:2 d6:0 d8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:1 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u5:0 u7:1] D[d4:0 d6:2 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:2 d6:0 d8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:0 d6:1 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u5:0 u7:1] D[d4:0 d6:2 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:1 d6:0 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:0 d6:1 d8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u5:0 u7:2] D[d4:0 d6:1 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:2 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:1 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:2 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u3:0 u5:1 u7:0] D[d4:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u5:0 u7:1] D[d4:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:2 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u3:2 u5:1 u7:0] D[d4:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u5:0 u7:1] D[d4:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:1 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u5:2 u7:0] D[d4:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u5:0 u7:2] D[d4:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]` ## C04 — word=UUUUDDDDD bites=(4,8) face=bite(4,8) apexes=[d5,d6,d7] 12 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `210` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d5:2 d6:1 d7:0 p4_8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:1 d7:2 p4_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d5:1 d6:2 d7:0 p4_8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:2 d7:1 p4_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d5:2 d6:1 d7:0 p4_8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:1 d7:2 p4_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d5:1 d6:2 d7:0 p4_8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d5:0 d6:2 d7:1 p4_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]` ## C05 — word=UUUDUUUDD bites=- face=root apexes=[d3,d7,d8] 13 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u5:2 u6:1] D[d3:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1 u5:1 u6:1] D[d3:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u6:0] D[d3:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u2:1 u4:2 u5:1 u6:0] D[d3:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u5:1 u6:2] D[d3:2 d7:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012020121 U[u0:2 u1:0 u2:1 u4:1 u5:2 u6:0] D[d3:1 d7:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u5:2 u6:1] D[d3:1 d7:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u5:2 u6:1] D[d3:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1 u5:1 u6:1] D[d3:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u6:0] D[d3:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u4:2 u5:1 u6:0] D[d3:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u5:1 u6:2] D[d3:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u4:1 u5:2 u6:0] D[d3:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u5:2 u6:1] D[d3:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2 u5:2 u6:2] D[d3:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]` ## C06 — word=UUUDUUDUD bites=- face=root apexes=[d3,d6,d8] 20 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=010120102 U[u0:2 u1:2 u2:2 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=010120201 U[u0:2 u1:2 u2:2 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=010121012 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u7:0] D[d3:1 d6:0 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=010210102 U[u0:2 u1:2 u2:1 u4:2 u5:2 u7:1] D[d3:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=010210201 U[u0:2 u1:2 u2:1 u4:2 u5:1 u7:2] D[d3:0 d6:1 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=010212021 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:1 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u5:0 u7:1] D[d3:0 d6:2 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012010212 U[u0:2 u1:0 u2:1 u4:2 u5:1 u7:0] D[d3:2 d6:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012020121 U[u0:2 u1:0 u2:1 u4:1 u5:2 u7:0] D[d3:1 d6:0 d8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=012120102 U[u0:2 u1:0 u2:0 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012120201 U[u0:2 u1:0 u2:0 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=012121012 U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012121021 U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u7:0] D[d3:1 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u4:2 u5:2 u7:1] D[d3:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u4:2 u5:1 u7:2] D[d3:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u4:0 u5:1 u7:0] D[d3:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u5:0 u7:1] D[d3:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u4:2 u5:1 u7:0] D[d3:2 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u4:1 u5:2 u7:0] D[d3:1 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]` ## C07 — word=UUUDUDDDD bites=(3,5) face=root apexes=[d6,d7,d8] 6 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d6:2 d7:0 d8:1 p3_5:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d6:1 d7:0 d8:2 p3_5:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d6:2 d7:0 d8:1 p3_5:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d6:1 d7:0 d8:2 p3_5:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]` ## C08 — word=UUUDUDDDD bites=(3,8) face=bite(3,8) apexes=[d5,d6,d7] 6 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `210` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d5:1 d6:2 d7:0 p3_8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d5:0 d6:2 d7:1 p3_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d5:1 d6:2 d7:0 p3_8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d5:0 d6:2 d7:1 p3_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]` ## C09 — word=UUUDDUDDD bites=(4,6) face=root apexes=[d3,d7,d8] 3 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d7:0 d8:2 p4_6:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d7:0 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]` ## C10 — word=UUUDDUDDD bites=(3,8) face=bite(3,8) apexes=[d4,d6,d7] 3 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `210` → canonical `012` · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d4:1 d6:2 d7:0 p3_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d4:1 d6:2 d7:0 p3_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]` ## C11 — word=UUDUUDUUD bites=- face=root apexes=[d2,d5,d8] 18 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101202 U[u0:2 u1:2 u3:2 u4:2 u6:1 u7:1] D[d2:2 d5:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010101212 U[u0:2 u1:2 u3:2 u4:2 u6:0 u7:0] D[d2:2 d5:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010102102 U[u0:2 u1:2 u3:2 u4:1 u6:2 u7:1] D[d2:2 d5:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121202 U[u0:2 u1:2 u3:0 u4:0 u6:1 u7:1] D[d2:2 d5:0 d8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:2 d5:0 d8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010201201 U[u0:2 u1:2 u3:1 u4:2 u6:1 u7:2] D[d2:1 d5:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202101 U[u0:2 u1:2 u3:1 u4:1 u6:2 u7:2] D[d2:1 d5:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202121 U[u0:2 u1:2 u3:1 u4:1 u6:0 u7:0] D[d2:1 d5:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010212101 U[u0:2 u1:2 u3:0 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=012012101 U[u0:2 u1:0 u3:2 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=012012121 U[u0:2 u1:0 u3:2 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=012021201 U[u0:2 u1:0 u3:1 u4:0 u6:1 u7:2] D[d2:1 d5:0 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=012101012 U[u0:2 u1:0 u3:2 u4:2 u6:2 u7:0] D[d2:0 d5:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012102021 U[u0:2 u1:0 u3:2 u4:1 u6:1 u7:0] D[d2:0 d5:1 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=012120102 U[u0:2 u1:0 u3:0 u4:1 u6:2 u7:1] D[d2:0 d5:2 d8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012120201 U[u0:2 u1:0 u3:0 u4:1 u6:1 u7:2] D[d2:0 d5:1 d8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=012121012 U[u0:2 u1:0 u3:0 u4:0 u6:2 u7:0] D[d2:0 d5:2 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2 u6:1 u7:1] D[d2:2 d5:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2 u6:0 u7:0] D[d2:2 d5:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u4:1 u6:2 u7:1] D[d2:2 d5:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0 u6:1 u7:1] D[d2:2 d5:0 d8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:2 d5:0 d8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u4:2 u6:1 u7:2] D[d2:1 d5:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1 u6:2 u7:2] D[d2:1 d5:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1 u6:0 u7:0] D[d2:1 d5:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012021201 U[u0:2 u1:0 u3:1 u4:0 u6:1 u7:2] D[d2:1 d5:0 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u4:2 u6:2 u7:0] D[d2:0 d5:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u4:1 u6:1 u7:0] D[d2:0 d5:1 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u4:1 u6:2 u7:1] D[d2:0 d5:2 d8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u4:1 u6:1 u7:2] D[d2:0 d5:1 d8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u4:0 u6:2 u7:0] D[d2:0 d5:2 d8:1]` ## C12 — word=UUDUUDDDD bites=(2,5) face=root apexes=[d6,d7,d8] 10 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d6:2 d7:0 d8:1 p2_5:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d6:1 d7:0 d8:2 p2_5:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d6:2 d7:0 d8:1 p2_5:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d6:1 d7:0 d8:2 p2_5:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d6:2 d7:0 d8:1 p2_5:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d6:1 d7:0 d8:2 p2_5:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d6:2 d7:0 d8:1 p2_5:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d6:1 d7:0 d8:2 p2_5:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d6:2 d7:0 d8:1 p2_5:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d6:1 d7:0 d8:2 p2_5:1]` ## C13 — word=UUDUDUDDD bites=(4,6) face=root apexes=[d2,d7,d8] 5 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d7:0 d8:1 p4_6:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d7:0 d8:1 p4_6:0]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d7:0 d8:2 p4_6:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]` -- face apexes (raw labels) `102` → canonical `012` · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d7:0 d8:1 p4_6:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d7:0 d8:1 p4_6:0]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d7:0 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d7:0 d8:2 p4_6:0]` ## C14 — word=UUDUDUDDD bites=(2,4) face=root apexes=[d6,d7,d8] 6 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d6:2 d7:0 d8:1 p2_4:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d6:1 d7:0 d8:2 p2_4:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d6:2 d7:0 d8:1 p2_4:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d6:1 d7:0 d8:2 p2_4:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d6:2 d7:0 d8:1 p2_4:0]` -- face apexes (raw labels) `102` → canonical `012` · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d6:1 d7:0 d8:2 p2_4:0]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d6:2 d7:0 d8:1 p2_4:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d6:1 d7:0 d8:2 p2_4:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d6:2 d7:0 d8:1 p2_4:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d6:1 d7:0 d8:2 p2_4:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d6:2 d7:0 d8:1 p2_4:0]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d6:1 d7:0 d8:2 p2_4:0]` ## C15 — word=UUDUDUDDD bites=(2,8) face=bite(2,8) apexes=[d4,d6,d7] 5 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `210` → canonical `012` · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d4:2 d6:1 d7:0 p2_8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:1 d7:2 p2_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d4:1 d6:2 d7:0 p2_8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d4:2 d6:1 d7:0 p2_8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:1 d7:2 p2_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d4:1 d6:2 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d4:0 d6:2 d7:1 p2_8:1]` ## C16 — word=UUDUDDUDD bites=(5,7) face=root apexes=[d2,d4,d8] 6 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d8:1 p5_7:0]` -- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]` -- face apexes (raw labels) `102` → canonical `012` · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]` -- face apexes (raw labels) `021` → canonical `012` · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d8:1 p5_7:0]` -- face apexes (raw labels) `012` → canonical `012` · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d8:2 p5_7:0]` -- face apexes (raw labels) `012` → canonical `012` · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d8:2 p5_7:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d8:1 p5_7:0]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d8:2 p5_7:0]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d8:1 p5_7:0]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d8:2 p5_7:0]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d8:2 p5_7:2]` ## C17 — word=UUDUDDUDD bites=(2,4) face=root apexes=[d5,d7,d8] 3 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d5:2 d7:0 d8:1 p2_4:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d5:1 d7:0 d8:2 p2_4:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d5:2 d7:0 d8:1 p2_4:0]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d5:2 d7:0 d8:1 p2_4:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d5:1 d7:0 d8:2 p2_4:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d5:2 d7:0 d8:1 p2_4:0]` ## C18 — word=UUDUDDUDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d7] 5 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `102` → canonical `012` · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d4:1 d5:0 d7:2 p2_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d4:1 d5:2 d7:0 p2_8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d4:2 d5:0 d7:1 p2_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d4:1 d5:0 d7:2 p2_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d4:1 d5:2 d7:0 p2_8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d4:2 d5:0 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d4:2 d5:1 d7:0 p2_8:1]` ## C19 — word=UUDUDDDUD bites=(6,8) face=root apexes=[d2,d4,d5] 3 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `210` → canonical `012` · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 p6_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 p6_8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 p6_8:2]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 p6_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 p6_8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 p6_8:2]` ## C20 — word=UUDUDDDUD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6] 10 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d4:2 d5:0 d6:1 p2_8:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d4:1 d5:0 d6:2 p2_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d4:1 d5:2 d6:0 p2_8:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:2 d6:1 p2_8:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d4:2 d5:0 d6:1 p2_8:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d4:1 d5:0 d6:2 p2_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d4:2 d5:0 d6:1 p2_8:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d4:1 d5:0 d6:2 p2_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d4:1 d5:2 d6:0 p2_8:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:2 d6:1 p2_8:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d4:2 d5:0 d6:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d4:1 d5:0 d6:2 p2_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d4:2 d5:1 d6:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d4:0 d5:1 d6:2 p2_8:1]` ## C21 — word=UUDDUUDDD bites=(3,6) face=root apexes=[d2,d7,d8] 9 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d7:0 d8:1 p3_6:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:0]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d7:0 d8:1 p3_6:0]` -- face apexes (raw labels) `102` → canonical `012` · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d7:0 d8:2 p3_6:1]` -- face apexes (raw labels) `102` → canonical `012` · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:0]` -- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d7:0 d8:2 p3_6:0]` -- face apexes (raw labels) `102` → canonical `012` · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d7:0 d8:1 p3_6:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d7:0 d8:1 p3_6:0]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d7:0 d8:1 p3_6:0]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d7:0 d8:2 p3_6:1]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d7:0 d8:2 p3_6:0]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d7:0 d8:2 p3_6:0]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]` ## C22 — word=UUDDUDUDD bites=(5,7) face=root apexes=[d2,d3,d8] 5 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d8:1 p5_7:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d8:1 p5_7:0]` -- face apexes (raw labels) `102` → canonical `012` · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d8:2 p5_7:2]` -- face apexes (raw labels) `102` → canonical `012` · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d8:2 p5_7:0]` -- face apexes (raw labels) `021` → canonical `012` · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d8:1 p5_7:0]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d8:1 p5_7:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d8:1 p5_7:0]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d8:2 p5_7:2]` +- face apex colours (canonical-rep edge order) `102`, realises `012` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d8:2 p5_7:0]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d8:1 p5_7:0]` ## C23 — word=UUDDUDUDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d7] 8 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `210` → canonical `012` · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d3:2 d5:1 d7:0 p2_8:2]` -- face apexes (raw labels) `012` → canonical `012` · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d3:0 d5:1 d7:2 p2_8:2]` -- face apexes (raw labels) `120` → canonical `012` · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d3:0 d5:2 d7:1 p2_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d3:2 d5:1 d7:0 p2_8:1]` -- face apexes (raw labels) `210` → canonical `012` · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:1 d7:0 p2_8:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:0 d7:1 p2_8:1]` -- face apexes (raw labels) `120` → canonical `012` · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d3:2 d5:1 d7:0 p2_8:2]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d3:0 d5:1 d7:2 p2_8:2]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d3:0 d5:2 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d3:2 d5:1 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `210`, realises `012` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:1 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:0 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `120`, realises `012` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]` ## C24 — word=UDUDUDUDD bites=(5,7) face=root apexes=[d1,d3,d8] 6 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d8:1 p5_7:0]` -- face apexes (raw labels) `201` → canonical `012` · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]` -- face apexes (raw labels) `021` → canonical `012` · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d8:1 p5_7:1]` -- face apexes (raw labels) `012` → canonical `012` · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d8:2 p5_7:2]` -- face apexes (raw labels) `021` → canonical `012` · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d8:1 p5_7:0]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d8:1 p5_7:0]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d8:1 p5_7:1]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d8:1 p5_7:1]` +- face apex colours (canonical-rep edge order) `012`, realises `012` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d8:2 p5_7:2]` +- face apex colours (canonical-rep edge order) `021`, realises `012` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d8:1 p5_7:0]` ## C25 — word=UDUDUDUDD bites=(3,5) face=root apexes=[d1,d7,d8] 3 colouring(s) with down-apex sequence `012`: -- face apexes (raw labels) `201` → canonical `012` · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d7:0 d8:1 p3_5:2]` -- face apexes (raw labels) `201` → canonical `012` · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d7:0 d8:1 p3_5:0]` -- face apexes (raw labels) `201` → canonical `012` · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d7:0 d8:1 p3_5:1]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d7:0 d8:1 p3_5:2]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d7:0 d8:1 p3_5:0]` +- face apex colours (canonical-rep edge order) `201`, realises `012` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d7:0 d8:1 p3_5:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.pdf index e51f2f5..9da7597 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/summary.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/summary.md index f2bca84..ec302e6 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/summary.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/summary.md @@ -1,6 +1,6 @@ # Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=3) -Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 3 singleton down-tooth apexes: **26 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry). +Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 3 singleton down-tooth apexes: **26 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in cyclic order off the un-deduped census (every cyclic orientation of each colouring), and reduce it modulo colour permutation (NOT dihedral symmetry). Reading off the census makes the recorded vocabulary orientation-honest; see `../kempe_sequence_orientation_note.md`. - Total Kempe-balanced colourings (mod colour permutation): **241**. - Distinct canonical down-apex sequences overall: **1**. diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.md index 99eee45..0191c68 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.md @@ -1,10 +1,10 @@ # Inner-face down-apex sequence `0000` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0. - Realised by **21** of 23 configs (M(T), inner face). -- **64** Kempe-balanced colourings (mod colour permutation) produce it. +- **64** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_0000.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. @@ -13,168 +13,168 @@ Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D 10 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=010120101 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=010210202 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012010202 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012020101 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012120101 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:1 d8:1]` ## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8] 5 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `1111` → canonical `0000` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:1 d8:1]` ## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8] 5 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `1111` → canonical `0000` · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:2 d8:2]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:1 d8:1]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010120202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010210101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:1 d8:1]` ## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8] 7 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=010102101 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=010201202 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012012101 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012021202 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012102101 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]` ## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8] 2 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:1 d8:1]` ## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8] 2 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:2 d6:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:1 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:2 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:1 d6:1 d8:1]` ## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8] 7 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=010202102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:2 d8:2]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012010201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:1 d8:1]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:1 d8:1]` -- face apexes (raw labels) `2222` → canonical `0000` · `A=012101201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:2 d8:2]` ## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7] 4 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=012010101 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:2 d7:2 p3_8:2]` -- face apexes (raw labels) `0000` → canonical `0000` · `A=012012121 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:1 d7:1 p3_8:1]` -- face apexes (raw labels) `0000` → canonical `0000` · `A=012021212 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:2 d7:2 p3_8:2]` +- face apex colours (canonical-rep edge order) `0000`, realises `0000` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:1 d7:1 p3_8:1]` +- face apex colours (canonical-rep edge order) `0000`, realises `0000` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:0 d7:0 p3_8:1]` ## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8] 4 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=010120101 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=010210202 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012010202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010120101 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010210202 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012010202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:1 d8:1]` ## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8] 4 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=010101201 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:2 d8:2]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=010202102 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:1 d8:1]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020102 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:1 d8:1]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:1 d8:1]` ## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8] 1 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u7:1] D[d2:1 d4:1 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1 u5:1 u7:1] D[d2:1 d4:1 d6:1 d8:1]` ## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8] 2 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=012020101 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:2 d8:2 p2_4:1]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:1 d8:1 p2_4:1]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:2 d8:2 p2_4:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:1 d8:1 p2_4:1]` ## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7] 2 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:1 d7:1 p2_8:1]` -- face apexes (raw labels) `0000` → canonical `0000` · `A=012021212 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:0 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:1 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `0000`, realises `0000` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:0 d7:0 p2_8:1]` ## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8] 1 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:1 d8:1 p3_5:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:1 d8:1 p3_5:1]` ## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7] 1 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:1 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:1 d7:1 p2_8:1]` ## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8] 1 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:1 d8:1 p4_6:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:1 d8:1 p4_6:1]` ## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7] 1 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `1111` → canonical `0000` · `A=012020202 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:1 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012020202 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:1 d7:1 p2_8:1]` ## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8] 1 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:2 d8:2 p3_5:0]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:2 d8:2 p3_5:0]` ## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8] 2 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `2222` → canonical `0000` · `A=012120101 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:2 d8:2 p1_3:0]` -- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:1 d8:1 p1_3:0]` +- face apex colours (canonical-rep edge order) `2222`, realises `0000` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:2 d8:2 p1_3:0]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:1 d8:1 p1_3:0]` ## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8] 1 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `1111` → canonical `0000` · `A=012120202 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:1 d8:1 p1_3:0]` +- face apex colours (canonical-rep edge order) `1111`, realises `0000` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:1 d8:1 p1_3:0]` ## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7] 1 colouring(s) with down-apex sequence `0000`: -- face apexes (raw labels) `0000` → canonical `0000` · `A=010212121 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:0 d7:0 p1_8:2]` +- face apex colours (canonical-rep edge order) `0000`, realises `0000` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:0 d7:0 p1_8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.pdf index 5f502cd..b034bd4 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0000.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.md index 92ff799..62a8197 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.md @@ -1,234 +1,307 @@ # Inner-face down-apex sequence `0011` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1. - Realised by **23** of 23 configs (M(T), inner face). -- **108** Kempe-balanced colourings (mod colour permutation) produce it. +- **181** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_0011.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8] -10 colouring(s) with down-apex sequence `0011`: +20 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=010120102 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=010210201 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012010201 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012020102 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012120102 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]` ## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8] -5 colouring(s) with down-apex sequence `0011`: +10 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `1122` → canonical `0011` · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]` ## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8] 15 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `1122` → canonical `0011` · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:1 d8:1]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:1 d8:1]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012012101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012021202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]` ## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8] -7 colouring(s) with down-apex sequence `0011`: +14 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=010102102 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=010201201 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012012102 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012021201 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012102102 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]` ## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8] -8 colouring(s) with down-apex sequence `0011`: +13 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `0022` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:1 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:2 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:1 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]` ## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8] -3 colouring(s) with down-apex sequence `0011`: +15 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `0011` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u7:2] D[d3:0 d5:0 d6:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u7:2] D[d3:0 d5:0 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]` ## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8] -13 colouring(s) with down-apex sequence `0011`: +17 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=010101202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=010121201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=010121202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=010202101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:2 d8:2]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=010212102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:1 d8:1]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012010102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:1 d8:1]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012010202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:2 d8:2]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:2 d8:2]` -- face apexes (raw labels) `2211` → canonical `0011` · `A=012101202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:2 d4:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]` ## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7] -4 colouring(s) with down-apex sequence `0011`: +8 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2200` → canonical `0011` · `A=012010121 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:0 d7:0 p3_8:2]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012012101 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:2 d7:2 p3_8:2]` -- face apexes (raw labels) `1100` → canonical `0011` · `A=012020212 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:0 d7:0 p3_8:1]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012021202 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:1 d7:1 p3_8:1]` +- face apex colours (canonical-rep edge order) `2200`, realises `0011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:0 d7:0 p3_8:2]` +- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d4:2 d5:1 d6:1 d7:2 p3_8:2]` +- face apex colours (canonical-rep edge order) `0110`, realises `0011` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1] D[d4:0 d5:1 d6:1 d7:0 p3_8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:2 d7:2 p3_8:2]` +- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d4:1 d5:2 d6:2 d7:1 p3_8:1]` +- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:0 d7:0 p3_8:1]` +- face apex colours (canonical-rep edge order) `0220`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1] D[d4:0 d5:2 d6:2 d7:0 p3_8:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:1 d7:1 p3_8:1]` ## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8] -10 colouring(s) with down-apex sequence `0011`: +13 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=010120102 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=010210201 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012010201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012101201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012101202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012102101 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012102102 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010102121 U[u0:2 u1:2 u3:2 u4:1 u6:0] D[d2:2 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010120102 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010201212 U[u0:2 u1:2 u3:1 u4:2 u6:0] D[d2:1 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010210201 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012010201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[d2:1 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:1 d8:1]` ## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8] -6 colouring(s) with down-apex sequence `0011`: +11 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:2 d8:2]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:2 d8:2]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:2 d8:2]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012121201 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u5:2 u6:1] D[d2:2 d4:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[d2:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:1 d8:1]` ## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8] -4 colouring(s) with down-apex sequence `0011`: +10 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:2 d6:1 d8:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:1 d6:2 d8:2]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u7:2] D[d2:1 d4:1 d6:2 d8:2]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u7:1] D[d2:0 d4:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:2 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=010102021 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=010120201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:2 d4:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=010201012 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:1 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=010210102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u7:2] D[d2:1 d4:1 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[d2:1 d4:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u7:1] D[d2:0 d4:0 d6:1 d8:1]` ## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `0011`: +4 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=012020102 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:1 d8:1 p2_4:1]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:2 d8:2 p2_4:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:1 d8:1 p2_4:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012020121 U[u0:2 u1:0 u3:1] D[d5:2 d6:0 d7:0 d8:2 p2_4:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:2 d8:2 p2_4:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d5:1 d6:0 d7:0 d8:1 p2_4:1]` ## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7] -2 colouring(s) with down-apex sequence `0011`: +4 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `1100` → canonical `0011` · `A=012020212 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:0 d7:0 p2_8:1]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012021202 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:1 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d4:1 d5:2 d6:2 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:0 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `0220`, realises `0011` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1] D[d4:0 d5:2 d6:2 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:1 d7:1 p2_8:1]` ## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8] -1 colouring(s) with down-apex sequence `0011`: +2 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:2 d8:2 p3_5:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:2 d8:2 p3_5:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d6:0 d7:0 d8:1 p3_5:1]` ## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7] -1 colouring(s) with down-apex sequence `0011`: +2 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `1100` → canonical `0011` · `A=012020212 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:0 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0011` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d3:1 d5:2 d6:2 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:0 d7:0 p2_8:1]` ## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8] 3 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `1122` → canonical `0011` · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:2 d8:2 p4_6:1]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:2 d8:2 p4_6:1]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:1 d8:1 p4_6:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:2 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:2 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:1 d8:1 p4_6:1]` ## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7] 3 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=012010202 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:1 d7:1 p2_8:1]` -- face apexes (raw labels) `2200` → canonical `0011` · `A=012010212 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:0 d7:0 p2_8:1]` -- face apexes (raw labels) `1100` → canonical `0011` · `A=012020212 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:0 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:1 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `2200`, realises `0011` in some orientation · `A=012010212 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:0 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:0 d7:0 p2_8:1]` ## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8] -1 colouring(s) with down-apex sequence `0011`: +2 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:1 d8:1 p3_5:0]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:1 d8:1 p3_5:0]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d6:0 d7:0 d8:2 p3_5:0]` ## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `0011`: +4 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=012120102 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:1 d8:1 p1_3:0]` -- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:2 d8:2 p1_3:0]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=012120102 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:1 d8:1 p1_3:0]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=012120121 U[u0:2 u2:0 u4:1] D[d5:2 d6:0 d7:0 d8:2 p1_3:0]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:2 d8:2 p1_3:0]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012120212 U[u0:2 u2:0 u4:1] D[d5:1 d6:0 d7:0 d8:1 p1_3:0]` ## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8] -2 colouring(s) with down-apex sequence `0011`: +3 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `0022` → canonical `0011` · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:2 d8:2 p4_6:1]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:1 d8:1 p4_6:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0011` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d7:0 d8:2 p4_6:0]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:2 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:1 d8:1 p4_6:1]` ## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `0011`: +2 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `1122` → canonical `0011` · `A=012120201 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:2 d8:2 p1_3:0]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:2 d8:2 p1_3:0]` +- face apex colours (canonical-rep edge order) `1001`, realises `0011` in some orientation · `A=012120212 U[u0:2 u2:0 u5:1] D[d4:1 d6:0 d7:0 d8:1 p1_3:0]` ## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7] 3 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `1122` → canonical `0011` · `A=010202101 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:2 d7:2 p1_8:2]` -- face apexes (raw labels) `1100` → canonical `0011` · `A=010202121 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:0 d7:0 p1_8:2]` -- face apexes (raw labels) `0022` → canonical `0011` · `A=010212101 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:2 d7:2 p1_8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0011` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:2 d7:2 p1_8:2]` +- face apex colours (canonical-rep edge order) `1100`, realises `0011` in some orientation · `A=010202121 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:0 d7:0 p1_8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0011` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:2 d7:2 p1_8:2]` ## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8] -2 colouring(s) with down-apex sequence `0011`: +3 colouring(s) with down-apex sequence `0011`: -- face apexes (raw labels) `2211` → canonical `0011` · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d8:1 p5_7:1]` -- face apexes (raw labels) `0011` → canonical `0011` · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d8:1 p5_7:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0011` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d8:1 p5_7:1]` +- face apex colours (canonical-rep edge order) `2112`, realises `0011` in some orientation · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d8:2 p5_7:0]` +- face apex colours (canonical-rep edge order) `0011`, realises `0011` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d8:1 p5_7:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.pdf index ce2e223..941edad 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.png index 278b64e..044d08e 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0011.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.md index 0cfcfa8..dabdcc5 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.md @@ -1,10 +1,10 @@ # Inner-face down-apex sequence `0101` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1. - Realised by **12** of 23 configs (M(T), inner face). -- **53** Kempe-balanced colourings (mod colour permutation) produce it. +- **53** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_0101.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. @@ -13,112 +13,112 @@ Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D 5 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `0202` → canonical `0101` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:2 d7:0 d8:2]` ## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8] 9 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `0202` → canonical `0101` · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=012121021 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:0 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:2 d7:0 d8:2]` ## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8] 11 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `2121` → canonical `0101` · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]` -- face apexes (raw labels) `1212` → canonical `0101` · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:1 d8:2]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:2 d6:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:1 d6:0 d8:1]` -- face apexes (raw labels) `2121` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:2 d8:1]` -- face apexes (raw labels) `1212` → canonical `0101` · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:1 d8:2]` -- face apexes (raw labels) `2121` → canonical `0101` · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:0 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]` ## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8] 3 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `0101` → canonical `0101` · `A=010120212 U[u0:2 u1:2 u2:2 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=010210121 U[u0:2 u1:2 u2:1 u5:2 u6:0] D[d3:0 d4:2 d7:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u2:0 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u5:2 u6:0] D[d3:0 d4:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]` ## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8] 5 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `0202` → canonical `0101` · `A=012101021 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012102012 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=012120121 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=012121021 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:2 d7:0 d8:2]` ## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8] 3 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `0202` → canonical `0101` · `A=012101021 U[u0:2 u1:0 u3:2 u5:2 u6:1] D[d2:0 d4:2 d7:0 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012102012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:0 d4:1 d7:0 d8:1]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u6:0] D[d2:0 d4:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u5:2 u6:1] D[d2:0 d4:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:0 d4:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u6:0] D[d2:0 d4:1 d7:0 d8:1]` ## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8] 8 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `2121` → canonical `0101` · `A=010102012 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:2 d8:1]` -- face apexes (raw labels) `2121` → canonical `0101` · `A=010102102 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:1 d6:2 d8:1]` -- face apexes (raw labels) `2121` → canonical `0101` · `A=010120102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:2 d4:1 d6:2 d8:1]` -- face apexes (raw labels) `1212` → canonical `0101` · `A=010201021 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:1 d8:2]` -- face apexes (raw labels) `1212` → canonical `0101` · `A=010201201 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:2 d6:1 d8:2]` -- face apexes (raw labels) `1212` → canonical `0101` · `A=010210201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]` -- face apexes (raw labels) `1212` → canonical `0101` · `A=012010201 U[u0:2 u1:0 u3:2 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u7:0] D[d2:0 d4:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010120102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:2 d4:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010210201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=012010201 U[u0:2 u1:0 u3:2 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u7:0] D[d2:0 d4:1 d6:0 d8:1]` ## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8] 1 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `0202` → canonical `0101` · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d7:0 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d7:0 d8:2 p4_6:1]` ## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7] 1 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `2020` → canonical `0101` · `A=012012012 U[u0:2 u1:0 u5:1] D[d3:2 d4:0 d6:2 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `2020`, realises `0101` in some orientation · `A=012012012 U[u0:2 u1:0 u5:1] D[d3:2 d4:0 d6:2 d7:0 p2_8:1]` ## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8] 3 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `0202` → canonical `0101` · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d7:0 d8:2 p4_6:0]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d7:0 d8:1 p4_6:0]` -- face apexes (raw labels) `0202` → canonical `0101` · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d7:0 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d7:0 d8:2 p4_6:0]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d7:0 d8:1 p4_6:0]` +- face apex colours (canonical-rep edge order) `0202`, realises `0101` in some orientation · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d7:0 d8:2 p4_6:1]` ## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7] 1 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `1212` → canonical `0101` · `A=010201201 U[u0:2 u2:1 u5:0] D[d3:1 d4:2 d6:1 d7:2 p1_8:2]` +- face apex colours (canonical-rep edge order) `1212`, realises `0101` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d3:1 d4:2 d6:1 d7:2 p1_8:2]` ## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8] 3 colouring(s) with down-apex sequence `0101`: -- face apexes (raw labels) `2121` → canonical `0101` · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d8:1 p5_7:0]` -- face apexes (raw labels) `2121` → canonical `0101` · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d8:1 p5_7:1]` -- face apexes (raw labels) `0101` → canonical `0101` · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d8:1 p5_7:0]` +- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d8:1 p5_7:0]` +- face apex colours (canonical-rep edge order) `2121`, realises `0101` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d8:1 p5_7:1]` +- face apex colours (canonical-rep edge order) `0101`, realises `0101` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d8:1 p5_7:0]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.pdf index b0ff0e4..60e414e 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0101.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.md index 53a8dfa..328bf58 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.md @@ -1,179 +1,307 @@ # Inner-face down-apex sequence `0110` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1. -- Realised by **19** of 23 configs (M(T), inner face). -- **73** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **23** of 23 configs (M(T), inner face). +- **181** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_0110.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8] -10 colouring(s) with down-apex sequence `0110`: +20 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u3:0 u4:2] D[d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u4:1] D[d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u4:1] D[d5:1 d6:0 d7:0 d8:1]` ## C01 — word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8] -5 colouring(s) with down-apex sequence `0110`: +10 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `1001` → canonical `0110` · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010120212 U[u0:2 u1:2 u2:2 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010210121 U[u0:2 u1:2 u2:1 u3:0 u5:2] D[d4:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u3:1 u5:1] D[d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012120212 U[u0:2 u1:0 u2:0 u3:0 u5:1] D[d4:1 d6:0 d7:0 d8:1]` + +## C02 — word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8] + +15 colouring(s) with down-apex sequence `0110`: + +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010120201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010210102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:2 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u3:0 u6:2] D[d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1 u3:1 u6:1] D[d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u3:0 u6:1] D[d4:0 d5:0 d7:1 d8:1]` ## C03 — word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8] -7 colouring(s) with down-apex sequence `0110`: +14 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012010121 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u5:0] D[d3:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1 u4:1 u5:1] D[d3:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]` ## C04 — word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8] -5 colouring(s) with down-apex sequence `0110`: +13 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:1 d5:0 d7:0 d8:1]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:2 d5:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:1 d5:0 d7:0 d8:1]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u6:0] D[d3:1 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u4:0 u6:2] D[d3:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:2 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u4:1 u6:1] D[d3:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:1 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u6:1] D[d3:0 d5:0 d7:1 d8:1]` ## C05 — word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8] -12 colouring(s) with down-apex sequence `0110`: +15 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2112` → canonical `0110` · `A=010102021 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]` -- face apexes (raw labels) `1221` → canonical `0110` · `A=010201012 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:2 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:0 d6:0 d8:1]` -- face apexes (raw labels) `2112` → canonical `0110` · `A=012010201 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:1 d6:1 d8:2]` -- face apexes (raw labels) `2112` → canonical `0110` · `A=012012021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:1 d8:2]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:0 d6:0 d8:2]` -- face apexes (raw labels) `1221` → canonical `0110` · `A=012020102 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:2 d6:2 d8:1]` -- face apexes (raw labels) `1221` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:2 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:0 d6:0 d8:1]` -- face apexes (raw labels) `2112` → canonical `0110` · `A=012102021 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u4:2 u7:0] D[d3:1 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u4:0 u7:2] D[d3:0 d5:0 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1 u4:1 u7:1] D[d3:1 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u4:0 u7:1] D[d3:0 d5:0 d6:1 d8:1]` ## C06 — word=UUUDDUUDD bites=- face=root apexes=[d3,d4,d7,d8] -4 colouring(s) with down-apex sequence `0110`: +17 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=012012021 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:0 d7:0 d8:2]` -- face apexes (raw labels) `2002` → canonical `0110` · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:2 d4:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:0 d7:0 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010121201 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010121202 U[u0:2 u1:2 u2:2 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=010212102 U[u0:2 u1:2 u2:1 u5:0 u6:2] D[d3:0 d4:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:2 d4:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:2 d4:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1 u5:1 u6:1] D[d3:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u2:0 u5:0 u6:1] D[d3:0 d4:0 d7:1 d8:1]` ## C07 — word=UUUDDDDDD bites=(3,8) face=bite(3,8) apexes=[d4,d5,d6,d7] -4 colouring(s) with down-apex sequence `0110`: +8 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2112` → canonical `0110` · `A=012010201 U[u0:2 u1:0 u2:1] D[d4:2 d5:1 d6:1 d7:2 p3_8:2]` -- face apexes (raw labels) `0110` → canonical `0110` · `A=012012021 U[u0:2 u1:0 u2:1] D[d4:0 d5:1 d6:1 d7:0 p3_8:2]` -- face apexes (raw labels) `1221` → canonical `0110` · `A=012020102 U[u0:2 u1:0 u2:1] D[d4:1 d5:2 d6:2 d7:1 p3_8:1]` -- face apexes (raw labels) `0220` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u2:1] D[d4:0 d5:2 d6:2 d7:0 p3_8:1]` +- face apex colours (canonical-rep edge order) `2200`, realises `0110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d4:2 d5:2 d6:0 d7:0 p3_8:2]` +- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d4:2 d5:1 d6:1 d7:2 p3_8:2]` +- face apex colours (canonical-rep edge order) `0110`, realises `0110` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1] D[d4:0 d5:1 d6:1 d7:0 p3_8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:2 d7:2 p3_8:2]` +- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d4:1 d5:2 d6:2 d7:1 p3_8:1]` +- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d4:1 d5:1 d6:0 d7:0 p3_8:1]` +- face apex colours (canonical-rep edge order) `0220`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1] D[d4:0 d5:2 d6:2 d7:0 p3_8:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d4:0 d5:0 d6:1 d7:1 p3_8:1]` ## C08 — word=UUDUUDUDD bites=- face=root apexes=[d2,d5,d7,d8] -3 colouring(s) with down-apex sequence `0110`: +13 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=010102121 U[u0:2 u1:2 u3:2 u4:1 u6:0] D[d2:2 d5:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=010201212 U[u0:2 u1:2 u3:1 u4:2 u6:0] D[d2:1 d5:0 d7:0 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[d2:1 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010102121 U[u0:2 u1:2 u3:2 u4:1 u6:0] D[d2:2 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010120102 U[u0:2 u1:2 u3:0 u4:1 u6:2] D[d2:2 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010201212 U[u0:2 u1:2 u3:1 u4:2 u6:0] D[d2:1 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010210201 U[u0:2 u1:2 u3:0 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012010201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1 u4:1 u6:1] D[d2:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[d2:1 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u4:0 u6:1] D[d2:0 d5:0 d7:1 d8:1]` ## C09 — word=UUDUDUUDD bites=- face=root apexes=[d2,d4,d7,d8] -5 colouring(s) with down-apex sequence `0110`: +11 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=010121021 U[u0:2 u1:2 u3:0 u5:2 u6:1] D[d2:2 d4:0 d7:0 d8:2]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=010212012 U[u0:2 u1:2 u3:0 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:0 d7:0 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[d2:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u6:1] D[d2:2 d4:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u5:2 u6:1] D[d2:2 d4:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u6:2] D[d2:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1 u5:1 u6:1] D[d2:1 d4:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[d2:1 d4:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u6:1] D[d2:0 d4:0 d7:1 d8:1]` ## C10 — word=UUDUDUDUD bites=- face=root apexes=[d2,d4,d6,d8] -6 colouring(s) with down-apex sequence `0110`: +10 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2112` → canonical `0110` · `A=010102021 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:1 d8:2]` -- face apexes (raw labels) `2112` → canonical `0110` · `A=010120201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:2 d4:1 d6:1 d8:2]` -- face apexes (raw labels) `1221` → canonical `0110` · `A=010201012 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:2 d8:1]` -- face apexes (raw labels) `1221` → canonical `0110` · `A=010210102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]` -- face apexes (raw labels) `1221` → canonical `0110` · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[d2:1 d4:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u5:0 u7:1] D[d2:2 d4:2 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=010102021 U[u0:2 u1:2 u3:2 u5:1 u7:0] D[d2:2 d4:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=010120201 U[u0:2 u1:2 u3:0 u5:1 u7:2] D[d2:2 d4:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=010201012 U[u0:2 u1:2 u3:1 u5:2 u7:0] D[d2:1 d4:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u5:0 u7:2] D[d2:1 d4:1 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=010210102 U[u0:2 u1:2 u3:0 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u7:2] D[d2:1 d4:1 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[d2:1 d4:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012121202 U[u0:2 u1:0 u3:0 u5:0 u7:1] D[d2:0 d4:0 d6:1 d8:1]` ## C11 — word=UUDUDDDDD bites=(2,4) face=root apexes=[d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `0110`: +4 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=012020121 U[u0:2 u1:0 u3:1] D[d5:2 d6:0 d7:0 d8:2 p2_4:1]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u3:1] D[d5:1 d6:0 d7:0 d8:1 p2_4:1]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d5:2 d6:2 d7:1 d8:1 p2_4:1]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012020121 U[u0:2 u1:0 u3:1] D[d5:2 d6:0 d7:0 d8:2 p2_4:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d5:1 d6:1 d7:2 d8:2 p2_4:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d5:1 d6:0 d7:0 d8:1 p2_4:1]` ## C12 — word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7] -2 colouring(s) with down-apex sequence `0110`: +4 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `1221` → canonical `0110` · `A=012020102 U[u0:2 u1:0 u3:1] D[d4:1 d5:2 d6:2 d7:1 p2_8:1]` -- face apexes (raw labels) `0220` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u3:1] D[d4:0 d5:2 d6:2 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d4:1 d5:2 d6:2 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d4:1 d5:1 d6:0 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `0220`, realises `0110` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1] D[d4:0 d5:2 d6:2 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d4:0 d5:0 d6:1 d7:1 p2_8:1]` ## C13 — word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8] -1 colouring(s) with down-apex sequence `0110`: +2 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `1001` → canonical `0110` · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d6:0 d7:0 d8:1 p3_5:1]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d6:1 d7:2 d8:2 p3_5:1]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d6:0 d7:0 d8:1 p3_5:1]` ## C14 — word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7] -1 colouring(s) with down-apex sequence `0110`: +2 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `1221` → canonical `0110` · `A=012020102 U[u0:2 u1:0 u4:1] D[d3:1 d5:2 d6:2 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `1221`, realises `0110` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d3:1 d5:2 d6:2 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d3:1 d5:1 d6:0 d7:0 p2_8:1]` + +## C15 — word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8] + +3 colouring(s) with down-apex sequence `0110`: + +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d7:2 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:2 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d7:1 d8:1 p4_6:1]` + +## C16 — word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7] + +3 colouring(s) with down-apex sequence `0110`: + +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:1 d7:1 p2_8:1]` +- face apex colours (canonical-rep edge order) `2200`, realises `0110` in some orientation · `A=012010212 U[u0:2 u1:0 u5:1] D[d3:2 d4:2 d6:0 d7:0 p2_8:1]` +- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d3:1 d4:1 d6:0 d7:0 p2_8:1]` ## C17 — word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8] -1 colouring(s) with down-apex sequence `0110`: +2 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d6:0 d7:0 d8:2 p3_5:0]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d6:2 d7:1 d8:1 p3_5:0]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d6:0 d7:0 d8:2 p3_5:0]` ## C18 — word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `0110`: +4 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=012120121 U[u0:2 u2:0 u4:1] D[d5:2 d6:0 d7:0 d8:2 p1_3:0]` -- face apexes (raw labels) `1001` → canonical `0110` · `A=012120212 U[u0:2 u2:0 u4:1] D[d5:1 d6:0 d7:0 d8:1 p1_3:0]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=012120102 U[u0:2 u2:0 u4:1] D[d5:2 d6:2 d7:1 d8:1 p1_3:0]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=012120121 U[u0:2 u2:0 u4:1] D[d5:2 d6:0 d7:0 d8:2 p1_3:0]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u2:0 u4:1] D[d5:1 d6:1 d7:2 d8:2 p1_3:0]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012120212 U[u0:2 u2:0 u4:1] D[d5:1 d6:0 d7:0 d8:1 p1_3:0]` ## C19 — word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8] -1 colouring(s) with down-apex sequence `0110`: +3 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2002` → canonical `0110` · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d7:0 d8:2 p4_6:0]` +- face apex colours (canonical-rep edge order) `2002`, realises `0110` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d7:0 d8:2 p4_6:0]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:2 d8:2 p4_6:1]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d7:1 d8:1 p4_6:1]` ## C20 — word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `0110`: +2 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `1001` → canonical `0110` · `A=012120212 U[u0:2 u2:0 u5:1] D[d4:1 d6:0 d7:0 d8:1 p1_3:0]` +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d4:1 d6:1 d7:2 d8:2 p1_3:0]` +- face apex colours (canonical-rep edge order) `1001`, realises `0110` in some orientation · `A=012120212 U[u0:2 u2:0 u5:1] D[d4:1 d6:0 d7:0 d8:1 p1_3:0]` + +## C21 — word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7] + +3 colouring(s) with down-apex sequence `0110`: + +- face apex colours (canonical-rep edge order) `1122`, realises `0110` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:2 d7:2 p1_8:2]` +- face apex colours (canonical-rep edge order) `1100`, realises `0110` in some orientation · `A=010202121 U[u0:2 u2:1 u5:0] D[d3:1 d4:1 d6:0 d7:0 p1_8:2]` +- face apex colours (canonical-rep edge order) `0022`, realises `0110` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d3:0 d4:0 d6:2 d7:2 p1_8:2]` ## C22 — word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8] -1 colouring(s) with down-apex sequence `0110`: +3 colouring(s) with down-apex sequence `0110`: -- face apexes (raw labels) `2112` → canonical `0110` · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d8:2 p5_7:0]` +- face apex colours (canonical-rep edge order) `2211`, realises `0110` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d8:1 p5_7:1]` +- face apex colours (canonical-rep edge order) `2112`, realises `0110` in some orientation · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d8:2 p5_7:0]` +- face apex colours (canonical-rep edge order) `0011`, realises `0110` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d8:1 p5_7:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.pdf index f282dea..e14d2f2 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.png index 48dd71c..89d0088 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/seq_0110.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/summary.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/summary.md index 274cf81..96d3471 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/summary.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m4/summary.md @@ -1,6 +1,6 @@ # Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=4) -Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 4 singleton down-tooth apexes: **23 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry). +Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 4 singleton down-tooth apexes: **23 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in cyclic order off the un-deduped census (every cyclic orientation of each colouring), and reduce it modulo colour permutation (NOT dihedral symmetry). Reading off the census makes the recorded vocabulary orientation-honest; see `../kempe_sequence_orientation_note.md`. - Total Kempe-balanced colourings (mod colour permutation): **298**. - Distinct canonical down-apex sequences overall: **4**. @@ -10,21 +10,20 @@ Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedr | sequence | colour multiset | #configs realising | #colourings | |---|---|---|---| | `0000` | 4 | 21 | 64 | -| `0011` | 2+2 | 23 | 108 | +| `0011` | 2+2 | 23 | 181 | | `0101` | 2+2 | 12 | 53 | -| `0110` | 2+2 | 19 | 73 | +| `0110` | 2+2 | 23 | 181 | Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 4 apexes (m is even) every count must be **even**, so the only admissible colour multisets are 2+2, 4. ## Step 4 — grouping configs by their set of unique down-apex sequences -The 23 configs fall into **4** groups by the set of canonical down-apex sequences they realise: +The 23 configs fall into **3** groups by the set of canonical down-apex sequences they realise: | #configs | set of down-apex sequences | config ids | |---|---|---| | 11 | { `0000`, `0011`, `0110` } | C00, C01, C03, C07, C11, C12, C13, C14, C17, C18, C20 | -| 6 | { `0000`, `0011`, `0101`, `0110` } | C04, C05, C06, C08, C09, C10 | -| 4 | { `0000`, `0011`, `0101` } | C02, C15, C16, C21 | +| 10 | { `0000`, `0011`, `0101`, `0110` } | C02, C04, C05, C06, C08, C09, C10, C15, C16, C21 | | 2 | { `0011`, `0101`, `0110` } | C19, C22 | ## Config atlas (ids) @@ -33,7 +32,7 @@ The 23 configs fall into **4** groups by the set of canonical down-apex sequence |---|---|---|---| | C00 | word=UUUUUDDDD bites=- face=root apexes=[d5,d6,d7,d8] | 30 | { `0000`, `0011`, `0110` } | | C01 | word=UUUUDUDDD bites=- face=root apexes=[d4,d6,d7,d8] | 15 | { `0000`, `0011`, `0110` } | -| C02 | word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8] | 25 | { `0000`, `0011`, `0101` } | +| C02 | word=UUUUDDUDD bites=- face=root apexes=[d4,d5,d7,d8] | 25 | { `0000`, `0011`, `0101`, `0110` } | | C03 | word=UUUDUUDDD bites=- face=root apexes=[d3,d6,d7,d8] | 21 | { `0000`, `0011`, `0110` } | | C04 | word=UUUDUDUDD bites=- face=root apexes=[d3,d5,d7,d8] | 24 | { `0000`, `0011`, `0101`, `0110` } | | C05 | word=UUUDUDDUD bites=- face=root apexes=[d3,d5,d6,d8] | 28 | { `0000`, `0011`, `0101`, `0110` } | @@ -46,13 +45,13 @@ The 23 configs fall into **4** groups by the set of canonical down-apex sequence | C12 | word=UUDUDDDDD bites=(2,8) face=bite(2,8) apexes=[d4,d5,d6,d7] | 6 | { `0000`, `0011`, `0110` } | | C13 | word=UUDDUDDDD bites=(3,5) face=root apexes=[d2,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } | | C14 | word=UUDDUDDDD bites=(2,8) face=bite(2,8) apexes=[d3,d5,d6,d7] | 3 | { `0000`, `0011`, `0110` } | -| C15 | word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8] | 5 | { `0000`, `0011`, `0101` } | -| C16 | word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101` } | +| C15 | word=UUDDDUDDD bites=(4,6) face=root apexes=[d2,d3,d7,d8] | 5 | { `0000`, `0011`, `0101`, `0110` } | +| C16 | word=UUDDDUDDD bites=(2,8) face=bite(2,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101`, `0110` } | | C17 | word=UDUDUDDDD bites=(3,5) face=root apexes=[d1,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } | | C18 | word=UDUDUDDDD bites=(1,3) face=root apexes=[d5,d6,d7,d8] | 6 | { `0000`, `0011`, `0110` } | | C19 | word=UDUDDUDDD bites=(4,6) face=root apexes=[d1,d3,d7,d8] | 6 | { `0011`, `0101`, `0110` } | | C20 | word=UDUDDUDDD bites=(1,3) face=root apexes=[d4,d6,d7,d8] | 3 | { `0000`, `0011`, `0110` } | -| C21 | word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101` } | +| C21 | word=UDUDDUDDD bites=(1,8) face=bite(1,8) apexes=[d3,d4,d6,d7] | 5 | { `0000`, `0011`, `0101`, `0110` } | | C22 | word=UDDUDDUDD bites=(5,7) face=root apexes=[d1,d2,d4,d8] | 6 | { `0011`, `0101`, `0110` } | ## Per-sequence notes diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.md index 4f2a72a..cd99a2a 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.md @@ -1,91 +1,222 @@ # Inner-face down-apex sequence `00012` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 3×colour0, 1×colour1, 1×colour2. - Realised by **10** of 10 configs (M(T), inner face). -- **30** Kempe-balanced colourings (mod colour permutation) produce it. +- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_00012.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8] -6 colouring(s) with down-apex sequence `00012`: +30 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `22201` → canonical `00012` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `22201` → canonical `00012` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` ## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8] -3 colouring(s) with down-apex sequence `00012`: +15 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `22201` → canonical `00012` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` ## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `00012`: +9 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `22201` → canonical `00012` · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `00012` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` ## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8] -5 colouring(s) with down-apex sequence `00012`: +25 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `22201` → canonical `00012` · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` ## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] -2 colouring(s) with down-apex sequence `00012`: +16 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `00012` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]` ## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] -2 colouring(s) with down-apex sequence `00012`: +8 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `00012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `00012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]` ## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] -1 colouring(s) with down-apex sequence `00012`: +12 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `00012` → canonical `00012` · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `22012`, realises `00012` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `00012` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `00012` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `11021`, realises `00012` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `02221`, realises `00012` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01112`, realises `00012` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `01112`, realises `00012` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `00012`, realises `00012` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]` ## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] -5 colouring(s) with down-apex sequence `00012`: +23 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `22201` → canonical `00012` · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `00012` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -2 colouring(s) with down-apex sequence `00012`: +12 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]` -- face apexes (raw labels) `11102` → canonical `00012` · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00012` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00012` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00012` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00012` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12011`, realises `00012` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00012` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `00012`: +11 colouring(s) with down-apex sequence `00012`: -- face apexes (raw labels) `22201` → canonical `00012` · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00012` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00012` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00012` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `00012` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `00012` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00012` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00012` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02111`, realises `00012` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `00012` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.pdf index 1c1c013..e1933fd 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.png index 3ded789..461cc66 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00012.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.md index 68be448..91958a4 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.md @@ -1,54 +1,114 @@ # Inner-face down-apex sequence `00102` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 3×colour0, 1×colour1, 1×colour2. -- Realised by **6** of 10 configs (M(T), inner face). -- **13** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **7** of 10 configs (M(T), inner face). +- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_00102.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] -6 colouring(s) with down-apex sequence `00102`: +12 colouring(s) with down-apex sequence `00102`: -- face apexes (raw labels) `00201` → canonical `00102` · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `00102` → canonical `00102` · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `00201` → canonical `00102` · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `00102` → canonical `00102` · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `00102` → canonical `00102` · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02122`, realises `00102` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` ## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] -2 colouring(s) with down-apex sequence `00102`: +8 colouring(s) with down-apex sequence `00102`: -- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `00102` → canonical `00102` · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02122`, realises `00102` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]` ## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `00102`: +14 colouring(s) with down-apex sequence `00102`: -- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21101`, realises `00102` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `00102` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `00102` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `00102` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12022`, realises `00102` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]` + +## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] + +16 colouring(s) with down-apex sequence `00102`: + +- face apex colours (canonical-rep edge order) `21202`, realises `00102` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `00102` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `20221`, realises `00102` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `20212`, realises `00102` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `00102` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `00102` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `00102` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `00102` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02212`, realises `00102` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `01121`, realises `00102` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `00102` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]` ## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] -2 colouring(s) with down-apex sequence `00102`: +4 colouring(s) with down-apex sequence `00102`: -- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `00102` → canonical `00102` · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02122`, realises `00102` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `00102` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `00102`: +6 colouring(s) with down-apex sequence `00102`: -- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `00102` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12101`, realises `00102` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12022`, realises `00102` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `00102`: +8 colouring(s) with down-apex sequence `00102`: -- face apexes (raw labels) `00201` → canonical `00102` · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `00102` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21011`, realises `00102` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21101`, realises `00102` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `00102` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02122`, realises `00102` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `00102` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `00102` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `00102` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.pdf index 1e22df0..52a2334 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.png index 0056fbd..f9e0aa2 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00102.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.md index 19784a3..73c6556 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.md @@ -1,86 +1,222 @@ # Inner-face down-apex sequence `00120` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 3×colour0, 1×colour1, 1×colour2. -- Realised by **9** of 10 configs (M(T), inner face). -- **30** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **10** of 10 configs (M(T), inner face). +- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_00120.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8] -6 colouring(s) with down-apex sequence `00120`: +30 colouring(s) with down-apex sequence `00120`: -- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `22102` → canonical `00120` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `22102` → canonical `00120` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` ## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8] -3 colouring(s) with down-apex sequence `00120`: +15 colouring(s) with down-apex sequence `00120`: -- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `22102` → canonical `00120` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` ## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `00120`: +9 colouring(s) with down-apex sequence `00120`: -- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `22102` → canonical `00120` · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `00120` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` ## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8] -5 colouring(s) with down-apex sequence `00120`: +25 colouring(s) with down-apex sequence `00120`: -- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `22102` → canonical `00120` · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` ## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] -2 colouring(s) with down-apex sequence `00120`: +16 colouring(s) with down-apex sequence `00120`: -- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `00120` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]` + +## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] + +8 colouring(s) with down-apex sequence `00120`: + +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `00120` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `00120` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]` ## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] -2 colouring(s) with down-apex sequence `00120`: +12 colouring(s) with down-apex sequence `00120`: -- face apexes (raw labels) `22012` → canonical `00120` · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]` -- face apexes (raw labels) `11021` → canonical `00120` · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `22012`, realises `00120` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `00120` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `00120` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `11021`, realises `00120` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `02221`, realises `00120` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01112`, realises `00120` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `01112`, realises `00120` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `00012`, realises `00120` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]` ## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] -5 colouring(s) with down-apex sequence `00120`: +23 colouring(s) with down-apex sequence `00120`: -- face apexes (raw labels) `22102` → canonical `00120` · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `22102` → canonical `00120` · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `00120` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -3 colouring(s) with down-apex sequence `00120`: +12 colouring(s) with down-apex sequence `00120`: -- face apexes (raw labels) `22102` → canonical `00120` · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` -- face apexes (raw labels) `11201` → canonical `00120` · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `00120` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `00120` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `00120` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `00120` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12011`, realises `00120` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `00120` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `00120`: +11 colouring(s) with down-apex sequence `00120`: -- face apexes (raw labels) `22102` → canonical `00120` · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `00120` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `00120` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `00120` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `00120` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `00120` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `00120` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `00120` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02111`, realises `00120` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `00120` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.pdf index c02c0f2..8e54a87 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.png index bb0e1d1..af8116f 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_00120.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.md index aeba829..e958456 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.md @@ -1,62 +1,114 @@ # Inner-face down-apex sequence `01002` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 3×colour0, 1×colour1, 1×colour2. - Realised by **7** of 10 configs (M(T), inner face). -- **16** Kempe-balanced colourings (mod colour permutation) produce it. +- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_01002.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `01002`: +12 colouring(s) with down-apex sequence `01002`: -- face apexes (raw labels) `01002` → canonical `01002` · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `02001` → canonical `01002` · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `01002` → canonical `01002` · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02122`, realises `01002` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` ## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `01002`: +8 colouring(s) with down-apex sequence `01002`: -- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `01002` → canonical `01002` · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `01002` → canonical `01002` · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02122`, realises `01002` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]` ## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] -2 colouring(s) with down-apex sequence `01002`: +14 colouring(s) with down-apex sequence `01002`: -- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]` -- face apexes (raw labels) `01002` → canonical `01002` · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01002` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `01002` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `01002` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01002` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12022`, realises `01002` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]` ## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] -5 colouring(s) with down-apex sequence `01002`: +16 colouring(s) with down-apex sequence `01002`: -- face apexes (raw labels) `20221` → canonical `01002` · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]` -- face apexes (raw labels) `10112` → canonical `01002` · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` -- face apexes (raw labels) `10112` → canonical `01002` · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` -- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]` -- face apexes (raw labels) `01002` → canonical `01002` · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `21202`, realises `01002` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01002` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `20221`, realises `01002` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `20212`, realises `01002` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `01002` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `01002` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `01002` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `01002` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02212`, realises `01002` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `01121`, realises `01002` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01002` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]` ## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] -1 colouring(s) with down-apex sequence `01002`: +4 colouring(s) with down-apex sequence `01002`: -- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `02122`, realises `01002` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01002` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `01002`: +6 colouring(s) with down-apex sequence `01002`: -- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01002` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12101`, realises `01002` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12022`, realises `01002` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `01002`: +8 colouring(s) with down-apex sequence `01002`: -- face apexes (raw labels) `02001` → canonical `01002` · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `01002` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21011`, realises `01002` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21101`, realises `01002` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `01002` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02122`, realises `01002` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01002` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `01002` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01002` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.pdf index 9d01ed1..5f725c2 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.png index ff68ae7..43c17ba 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01002.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.md index 23416b2..6ea2d79 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.md @@ -1,43 +1,114 @@ # Inner-face down-apex sequence `01020` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 3×colour0, 1×colour1, 1×colour2. -- Realised by **4** of 10 configs (M(T), inner face). -- **12** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **7** of 10 configs (M(T), inner face). +- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_01020.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] + +12 colouring(s) with down-apex sequence `01020`: + +- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02122`, realises `01020` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` + +## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] + +8 colouring(s) with down-apex sequence `01020`: + +- face apex colours (canonical-rep edge order) `02122`, realises `01020` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]` + ## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] -3 colouring(s) with down-apex sequence `01020`: +14 colouring(s) with down-apex sequence `01020`: -- face apexes (raw labels) `21202` → canonical `01020` · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `12101` → canonical `01020` · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `12101` → canonical `01020` · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21101`, realises `01020` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `01020` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `01020` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01020` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12022`, realises `01020` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]` ## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] -6 colouring(s) with down-apex sequence `01020`: +16 colouring(s) with down-apex sequence `01020`: -- face apexes (raw labels) `21202` → canonical `01020` · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]` -- face apexes (raw labels) `20212` → canonical `01020` · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]` -- face apexes (raw labels) `12101` → canonical `01020` · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` -- face apexes (raw labels) `10121` → canonical `01020` · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` -- face apexes (raw labels) `12101` → canonical `01020` · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` -- face apexes (raw labels) `10121` → canonical `01020` · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `01020` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01020` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `20221`, realises `01020` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `20212`, realises `01020` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `01020` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `01020` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `01020` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `01020` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02212`, realises `01020` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `01121`, realises `01020` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01020` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]` + +## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] + +4 colouring(s) with down-apex sequence `01020`: + +- face apex colours (canonical-rep edge order) `02122`, realises `01020` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01020` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -2 colouring(s) with down-apex sequence `01020`: +6 colouring(s) with down-apex sequence `01020`: -- face apexes (raw labels) `12101` → canonical `01020` · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `12101` → canonical `01020` · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01020` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12101`, realises `01020` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12022`, realises `01020` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `01020`: +8 colouring(s) with down-apex sequence `01020`: -- face apexes (raw labels) `21202` → canonical `01020` · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21202`, realises `01020` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21011`, realises `01020` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21101`, realises `01020` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `01020` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02122`, realises `01020` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01020` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `01020` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01020` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.pdf index 2c18c1a..eb798fc 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.png index 105ea44..21c127f 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01020.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.md index 706d48a..5d7f47e 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.md @@ -1,89 +1,222 @@ # Inner-face down-apex sequence `01112` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 1×colour0, 3×colour1, 1×colour2. -- Realised by **9** of 10 configs (M(T), inner face). -- **33** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **10** of 10 configs (M(T), inner face). +- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_01112.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8] -6 colouring(s) with down-apex sequence `01112`: +30 colouring(s) with down-apex sequence `01112`: -- face apexes (raw labels) `20001` → canonical `01112` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `20001` → canonical `01112` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `20001` → canonical `01112` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` ## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8] -3 colouring(s) with down-apex sequence `01112`: +15 colouring(s) with down-apex sequence `01112`: -- face apexes (raw labels) `20001` → canonical `01112` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `20001` → canonical `01112` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` + +## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] + +9 colouring(s) with down-apex sequence `01112`: + +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `01112` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` ## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8] -5 colouring(s) with down-apex sequence `01112`: +25 colouring(s) with down-apex sequence `01112`: -- face apexes (raw labels) `20001` → canonical `01112` · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` ## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `01112`: +16 colouring(s) with down-apex sequence `01112`: -- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `01112` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]` ## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] -3 colouring(s) with down-apex sequence `01112`: +8 colouring(s) with down-apex sequence `01112`: -- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `01112` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `01112` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]` ## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] -6 colouring(s) with down-apex sequence `01112`: +12 colouring(s) with down-apex sequence `01112`: -- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` -- face apexes (raw labels) `02221` → canonical `01112` · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]` -- face apexes (raw labels) `01112` → canonical `01112` · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]` -- face apexes (raw labels) `01112` → canonical `01112` · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `22012`, realises `01112` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `01112` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `01112` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `11021`, realises `01112` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `02221`, realises `01112` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01112`, realises `01112` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `01112`, realises `01112` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `00012`, realises `01112` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]` ## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] -4 colouring(s) with down-apex sequence `01112`: +23 colouring(s) with down-apex sequence `01112`: -- face apexes (raw labels) `20001` → canonical `01112` · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `01112` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -2 colouring(s) with down-apex sequence `01112`: +12 colouring(s) with down-apex sequence `01112`: -- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]` -- face apexes (raw labels) `10002` → canonical `01112` · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01112` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01112` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01112` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01112` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12011`, realises `01112` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01112` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `01112`: +11 colouring(s) with down-apex sequence `01112`: -- face apexes (raw labels) `20001` → canonical `01112` · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01112` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01112` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01112` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `01112` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `01112` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01112` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01112` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02111`, realises `01112` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `01112` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.pdf index 45660c2..0cb0fd6 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.png index 8b9e38f..981200e 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01112.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.md index 0742164..e8bfa2d 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.md @@ -1,43 +1,114 @@ # Inner-face down-apex sequence `01121` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 1×colour0, 3×colour1, 1×colour2. -- Realised by **4** of 10 configs (M(T), inner face). -- **12** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **7** of 10 configs (M(T), inner face). +- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_01121.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] + +12 colouring(s) with down-apex sequence `01121`: + +- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02122`, realises `01121` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` + +## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] + +8 colouring(s) with down-apex sequence `01121`: + +- face apex colours (canonical-rep edge order) `02122`, realises `01121` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]` + ## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] -5 colouring(s) with down-apex sequence `01121`: +14 colouring(s) with down-apex sequence `01121`: -- face apexes (raw labels) `21101` → canonical `01121` · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `21101` → canonical `01121` · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]` -- face apexes (raw labels) `12202` → canonical `01121` · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `12202` → canonical `01121` · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` -- face apexes (raw labels) `12202` → canonical `01121` · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01121` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `01121` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `01121` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01121` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12022`, realises `01121` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]` ## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] -5 colouring(s) with down-apex sequence `01121`: +16 colouring(s) with down-apex sequence `01121`: -- face apexes (raw labels) `21101` → canonical `01121` · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]` -- face apexes (raw labels) `12202` → canonical `01121` · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` -- face apexes (raw labels) `12202` → canonical `01121` · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` -- face apexes (raw labels) `02212` → canonical `01121` · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]` -- face apexes (raw labels) `01121` → canonical `01121` · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `01121` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01121` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `20221`, realises `01121` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `20212`, realises `01121` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `01121` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `01121` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `01121` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `01121` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02212`, realises `01121` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `01121`, realises `01121` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01121` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]` + +## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] + +4 colouring(s) with down-apex sequence `01121`: + +- face apex colours (canonical-rep edge order) `02122`, realises `01121` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01121` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `01121`: +6 colouring(s) with down-apex sequence `01121`: -- face apexes (raw labels) `12202` → canonical `01121` · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01121` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12101`, realises `01121` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12022`, realises `01121` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `01121`: +8 colouring(s) with down-apex sequence `01121`: -- face apexes (raw labels) `21101` → canonical `01121` · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `01121` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21011`, realises `01121` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21101`, realises `01121` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `01121` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02122`, realises `01121` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01121` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `01121` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01121` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.pdf index 88ad121..ac2d36a 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.png index 8d30b54..2dd6ed8 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01121.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.md index bbead4d..ab404e6 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.md @@ -1,88 +1,222 @@ # Inner-face down-apex sequence `01200` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 3×colour0, 1×colour1, 1×colour2. -- Realised by **9** of 10 configs (M(T), inner face). -- **32** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **10** of 10 configs (M(T), inner face). +- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_01200.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8] -6 colouring(s) with down-apex sequence `01200`: +30 colouring(s) with down-apex sequence `01200`: -- face apexes (raw labels) `20122` → canonical `01200` · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `20122` → canonical `01200` · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `20122` → canonical `01200` · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` ## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8] -3 colouring(s) with down-apex sequence `01200`: +15 colouring(s) with down-apex sequence `01200`: -- face apexes (raw labels) `20122` → canonical `01200` · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `20122` → canonical `01200` · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` + +## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] + +9 colouring(s) with down-apex sequence `01200`: + +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `01200` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` ## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8] -5 colouring(s) with down-apex sequence `01200`: +25 colouring(s) with down-apex sequence `01200`: -- face apexes (raw labels) `20122` → canonical `01200` · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `20122` → canonical `01200` · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` ## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `01200`: +16 colouring(s) with down-apex sequence `01200`: -- face apexes (raw labels) `20122` → canonical `01200` · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `01200` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]` ## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] -2 colouring(s) with down-apex sequence `01200`: +8 colouring(s) with down-apex sequence `01200`: -- face apexes (raw labels) `21022` → canonical `01200` · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `12011` → canonical `01200` · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `01200` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `01200` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]` ## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] -2 colouring(s) with down-apex sequence `01200`: +12 colouring(s) with down-apex sequence `01200`: -- face apexes (raw labels) `21022` → canonical `01200` · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]` -- face apexes (raw labels) `12011` → canonical `01200` · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `22012`, realises `01200` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `01200` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `01200` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `11021`, realises `01200` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `02221`, realises `01200` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01112`, realises `01200` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `01112`, realises `01200` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `00012`, realises `01200` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]` ## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] -4 colouring(s) with down-apex sequence `01200`: +23 colouring(s) with down-apex sequence `01200`: -- face apexes (raw labels) `20122` → canonical `01200` · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `20122` → canonical `01200` · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `01200` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -3 colouring(s) with down-apex sequence `01200`: +12 colouring(s) with down-apex sequence `01200`: -- face apexes (raw labels) `20122` → canonical `01200` · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `10211` → canonical `01200` · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]` -- face apexes (raw labels) `12011` → canonical `01200` · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01200` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01200` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01200` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01200` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12011`, realises `01200` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01200` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -4 colouring(s) with down-apex sequence `01200`: +11 colouring(s) with down-apex sequence `01200`: -- face apexes (raw labels) `20122` → canonical `01200` · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `21022` → canonical `01200` · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `21022` → canonical `01200` · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `20122` → canonical `01200` · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01200` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01200` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01200` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `01200` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `01200` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01200` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01200` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02111`, realises `01200` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `01200` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.pdf index df5fe2e..c2b0b3b 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.png index 169433a..c16cc8b 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01200.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.md index 8362125..00e272d 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.md @@ -1,56 +1,114 @@ # Inner-face down-apex sequence `01211` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 1×colour0, 3×colour1, 1×colour2. -- Realised by **6** of 10 configs (M(T), inner face). -- **15** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **7** of 10 configs (M(T), inner face). +- **68** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_01211.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `01211`: +12 colouring(s) with down-apex sequence `01211`: -- face apexes (raw labels) `01211` → canonical `01211` · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `02122` → canonical `01211` · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `01211` → canonical `01211` · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=010120102 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=010120121 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=010121012 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=010121021 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02122`, realises `01211` in some orientation · `A=010210201 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=010210212 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=010212012 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=010212021 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` ## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `01211`: +8 colouring(s) with down-apex sequence `01211`: -- face apexes (raw labels) `02122` → canonical `01211` · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `01211` → canonical `01211` · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `01211` → canonical `01211` · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02122`, realises `01211` in some orientation · `A=012101201 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012102102 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]` ## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] -3 colouring(s) with down-apex sequence `01211`: +14 colouring(s) with down-apex sequence `01211`: -- face apexes (raw labels) `21011` → canonical `01211` · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `12022` → canonical `01211` · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `01211` → canonical `01211` · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21101`, realises `01211` in some orientation · `A=010102012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `01211` in some orientation · `A=010102102 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `01211` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01211` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=010201021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12022`, realises `01211` in some orientation · `A=010201201 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u6:0] D[d2:0 d4:1 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]` + +## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] + +16 colouring(s) with down-apex sequence `01211`: + +- face apex colours (canonical-rep edge order) `21202`, realises `01211` in some orientation · `A=010120121 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `21101`, realises `01211` in some orientation · `A=010120212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:1 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `20221`, realises `01211` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `20212`, realises `01211` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=010210121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=010210212 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `01211` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `01211` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=012010121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:2 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=012010212 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:2 d5:1 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `10121`, realises `01211` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10112`, realises `01211` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02212`, realises `01211` in some orientation · `A=012101021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `01121`, realises `01211` in some orientation · `A=012102012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01002`, realises `01211` in some orientation · `A=012102121 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:0 d6:0 d8:2]` ## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] -1 colouring(s) with down-apex sequence `01211`: +4 colouring(s) with down-apex sequence `01211`: -- face apexes (raw labels) `02122` → canonical `01211` · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02122`, realises `01211` in some orientation · `A=012101201 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00102`, realises `01211` in some orientation · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -1 colouring(s) with down-apex sequence `01211`: +6 colouring(s) with down-apex sequence `01211`: -- face apexes (raw labels) `12022` → canonical `01211` · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `12202`, realises `01211` in some orientation · `A=012010121 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=012010212 U[u0:2 u1:0 u4:2 u6:0] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12101`, realises `01211` in some orientation · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12022`, realises `01211` in some orientation · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u1:0 u4:2 u6:0] D[d2:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -4 colouring(s) with down-apex sequence `01211`: +8 colouring(s) with down-apex sequence `01211`: -- face apexes (raw labels) `21011` → canonical `01211` · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `21011` → canonical `01211` · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `02122` → canonical `01211` · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `01211` → canonical `01211` · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21202`, realises `01211` in some orientation · `A=010201021 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `21011`, realises `01211` in some orientation · `A=010201202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `21101`, realises `01211` in some orientation · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21011`, realises `01211` in some orientation · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02122`, realises `01211` in some orientation · `A=012010201 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01211`, realises `01211` in some orientation · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02001`, realises `01211` in some orientation · `A=012101212 U[u0:2 u2:0 u4:2 u6:0] D[d1:0 d3:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `00201`, realises `01211` in some orientation · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.pdf index be26ac7..36f73a3 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.png index 5cf99bd..3c919b7 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01211.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.md index c2904db..0c37318 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.md @@ -1,97 +1,222 @@ # Inner-face down-apex sequence `01222` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 1×colour0, 1×colour1, 3×colour2. - Realised by **10** of 10 configs (M(T), inner face). -- **36** Kempe-balanced colourings (mod colour permutation) produce it. +- **161** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_01222.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8] -6 colouring(s) with down-apex sequence `01222`: +30 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `20111` → canonical `01222` · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `20111` → canonical `01222` · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `20111` → canonical `01222` · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010102012 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010102021 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010102101 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010102102 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010102121 U[u0:2 u1:2 u2:2 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010201012 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010201021 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010201201 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010201202 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010201212 U[u0:2 u1:2 u2:1 u3:1] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u3:1] D[d4:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` ## C01 — word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8] -3 colouring(s) with down-apex sequence `01222`: +15 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `20111` → canonical `01222` · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `20111` → canonical `01222` · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010101201 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010101202 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010101212 U[u0:2 u1:2 u2:2 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010202101 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010202102 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010202121 U[u0:2 u1:2 u2:1 u4:1] D[d3:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` ## C02 — word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `01222`: +9 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `01222` → canonical `01222` · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `02111` → canonical `01222` · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `01222` → canonical `01222` · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u2:2 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=010120101 U[u0:2 u1:2 u2:2 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u2:1 u5:1] D[d3:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `01222` in some orientation · `A=010210202 U[u0:2 u1:2 u2:1 u5:1] D[d3:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` ## C03 — word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8] -5 colouring(s) with down-apex sequence `01222`: +25 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `20111` → canonical `01222` · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `20111` → canonical `01222` · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010101202 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010101212 U[u0:2 u1:2 u3:2 u4:2] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010121012 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010121021 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u4:0] D[d2:2 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010202101 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010202121 U[u0:2 u1:2 u3:1 u4:1] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010212012 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010212021 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=012012012 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=012012021 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u4:0] D[d2:1 d5:0 d6:0 d7:0 d8:2]` ## C04 — word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] -6 colouring(s) with down-apex sequence `01222`: +16 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `20111` → canonical `01222` · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `02111` → canonical `01222` · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `01222` → canonical `01222` · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `01222` → canonical `01222` · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u3:2 u5:2] D[d2:2 d4:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010121201 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010121202 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u5:0] D[d2:2 d4:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u5:1] D[d2:1 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010212101 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010212102 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=012012101 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=012012102 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u5:0] D[d2:1 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `01222` in some orientation · `A=012101202 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012102101 U[u0:2 u1:0 u3:2 u5:0] D[d2:0 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]` ## C05 — word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `01222`: +8 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `01222` → canonical `01222` · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `01222` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u6:2] D[d2:2 d4:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u6:0] D[d2:2 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `01222` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u3:1 u6:1] D[d2:1 d4:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u6:0] D[d2:1 d4:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]` ## C06 — word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] -1 colouring(s) with down-apex sequence `01222`: +12 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `01222` → canonical `01222` · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `22012`, realises `01222` in some orientation · `A=010101201 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:2 d5:0 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `01222` in some orientation · `A=010102101 U[u0:2 u1:2 u3:2 u7:2] D[d2:2 d4:1 d5:0 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u3:0 u7:0] D[d2:2 d4:0 d5:0 d6:0 d8:1]` +- face apex colours (canonical-rep edge order) `12011`, realises `01222` in some orientation · `A=010201202 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:2 d5:0 d6:1 d8:1]` +- face apex colours (canonical-rep edge order) `11021`, realises `01222` in some orientation · `A=010202102 U[u0:2 u1:2 u3:1 u7:1] D[d2:1 d4:1 d5:0 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u3:0 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=012012121 U[u0:2 u1:0 u3:2 u7:0] D[d2:1 d4:0 d5:0 d6:0 d8:2]` +- face apex colours (canonical-rep edge order) `02221`, realises `01222` in some orientation · `A=012101012 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:2 d5:2 d6:2 d8:1]` +- face apex colours (canonical-rep edge order) `01112`, realises `01222` in some orientation · `A=012102021 U[u0:2 u1:0 u3:2 u7:0] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]` +- face apex colours (canonical-rep edge order) `01112`, realises `01222` in some orientation · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- face apex colours (canonical-rep edge order) `00012`, realises `01222` in some orientation · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]` ## C07 — word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] -5 colouring(s) with down-apex sequence `01222`: +23 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `20111` → canonical `01222` · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `20111` → canonical `01222` · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `02111` → canonical `01222` · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010101021 U[u0:2 u1:2 u4:2 u5:2] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010102012 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010120212 U[u0:2 u1:2 u4:1 u5:1] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010121201 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010121202 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u5:0] D[d2:2 d3:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010201021 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010202012 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u5:1] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010210121 U[u0:2 u1:2 u4:2 u5:2] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010212101 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010212102 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u5:0] D[d2:1 d3:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `02111`, realises `01222` in some orientation · `A=012101202 U[u0:2 u1:0 u4:2 u5:0] D[d2:0 d3:2 d6:1 d7:1 d8:1]` ## C08 — word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] -2 colouring(s) with down-apex sequence `01222`: +12 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `20111` → canonical `01222` · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]` -- face apexes (raw labels) `10222` → canonical `01222` · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u1:2 u4:2 u6:2] D[d2:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010102021 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010120201 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010120202 U[u0:2 u1:2 u4:1 u6:1] D[d2:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u1:2 u4:0 u6:0] D[d2:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=010201012 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `11102`, realises `01222` in some orientation · `A=010202021 U[u0:2 u1:2 u4:1 u6:1] D[d2:1 d3:1 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `10222`, realises `01222` in some orientation · `A=010210101 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `10211`, realises `01222` in some orientation · `A=010210102 U[u0:2 u1:2 u4:2 u6:2] D[d2:1 d3:0 d5:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `10002`, realises `01222` in some orientation · `A=010212121 U[u0:2 u1:2 u4:0 u6:0] D[d2:1 d3:0 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `12011`, realises `01222` in some orientation · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `11201`, realises `01222` in some orientation · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` ## C09 — word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] -4 colouring(s) with down-apex sequence `01222`: +11 colouring(s) with down-apex sequence `01222`: -- face apexes (raw labels) `20111` → canonical `01222` · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` -- face apexes (raw labels) `20111` → canonical `01222` · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` -- face apexes (raw labels) `02111` → canonical `01222` · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]` -- face apexes (raw labels) `01222` → canonical `01222` · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `22201`, realises `01222` in some orientation · `A=010101012 U[u0:2 u2:2 u4:2 u6:2] D[d1:2 d3:2 d5:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `22102`, realises `01222` in some orientation · `A=010102021 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:2 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010120201 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010120202 U[u0:2 u2:2 u4:1 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `20001`, realises `01222` in some orientation · `A=010121212 U[u0:2 u2:2 u4:0 u6:0] D[d1:2 d3:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `21022`, realises `01222` in some orientation · `A=010201201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `21022`, realises `01222` in some orientation · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20122`, realises `01222` in some orientation · `A=010210201 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `20111`, realises `01222` in some orientation · `A=010210202 U[u0:2 u2:1 u4:2 u6:1] D[d1:2 d3:0 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `02111`, realises `01222` in some orientation · `A=012010202 U[u0:2 u2:1 u4:2 u6:1] D[d1:0 d3:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `01222`, realises `01222` in some orientation · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.pdf index 4f39086..584d8a0 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.png index 1b854cb..d7e1771 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/seq_01222.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/summary.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/summary.md index e3e7656..b02bb94 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/summary.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m5/summary.md @@ -1,6 +1,6 @@ # Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=5) -Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 5 singleton down-tooth apexes: **10 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry). +Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 5 singleton down-tooth apexes: **10 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in cyclic order off the un-deduped census (every cyclic orientation of each colouring), and reduce it modulo colour permutation (NOT dihedral symmetry). Reading off the census makes the recorded vocabulary orientation-honest; see `../kempe_sequence_orientation_note.md`. - Total Kempe-balanced colourings (mod colour permutation): **229**. - Distinct canonical down-apex sequences overall: **10**. @@ -9,31 +9,27 @@ Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedr | sequence | colour multiset | #configs realising | #colourings | |---|---|---|---| -| `00012` | 3+1+1 | 10 | 30 | -| `00102` | 3+1+1 | 6 | 13 | -| `00120` | 3+1+1 | 9 | 30 | -| `01002` | 3+1+1 | 7 | 16 | -| `01020` | 3+1+1 | 4 | 12 | -| `01112` | 3+1+1 | 9 | 33 | -| `01121` | 3+1+1 | 4 | 12 | -| `01200` | 3+1+1 | 9 | 32 | -| `01211` | 3+1+1 | 6 | 15 | -| `01222` | 3+1+1 | 10 | 36 | +| `00012` | 3+1+1 | 10 | 161 | +| `00102` | 3+1+1 | 7 | 68 | +| `00120` | 3+1+1 | 10 | 161 | +| `01002` | 3+1+1 | 7 | 68 | +| `01020` | 3+1+1 | 7 | 68 | +| `01112` | 3+1+1 | 10 | 161 | +| `01121` | 3+1+1 | 7 | 68 | +| `01200` | 3+1+1 | 10 | 161 | +| `01211` | 3+1+1 | 7 | 68 | +| `01222` | 3+1+1 | 10 | 161 | Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 5 apexes (m is odd) every count must be **odd**, so the only admissible colour multisets are 3+1+1. ## Step 4 — grouping configs by their set of unique down-apex sequences -The 10 configs fall into **6** groups by the set of canonical down-apex sequences they realise: +The 10 configs fall into **2** groups by the set of canonical down-apex sequences they realise: | #configs | set of down-apex sequences | config ids | |---|---|---| +| 7 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | C02, C04, C05, C06, C07, C08, C09 | | 3 | { `00012`, `00120`, `01112`, `01200`, `01222` } | C00, C01, C03 | -| 2 | { `00012`, `00102`, `00120`, `01002`, `01112`, `01200`, `01211`, `01222` } | C04, C07 | -| 2 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | C08, C09 | -| 1 | { `00012`, `00102`, `00120`, `01002`, `01211`, `01222` } | C02 | -| 1 | { `00012`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01222` } | C06 | -| 1 | { `00012`, `00102`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | C05 | ## Config atlas (ids) @@ -41,12 +37,12 @@ The 10 configs fall into **6** groups by the set of canonical down-apex sequence |---|---|---|---| | C00 | word=UUUUDDDDD bites=- face=root apexes=[d4,d5,d6,d7,d8] | 30 | { `00012`, `00120`, `01112`, `01200`, `01222` } | | C01 | word=UUUDUDDDD bites=- face=root apexes=[d3,d5,d6,d7,d8] | 15 | { `00012`, `00120`, `01112`, `01200`, `01222` } | -| C02 | word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] | 21 | { `00012`, `00102`, `00120`, `01002`, `01211`, `01222` } | +| C02 | word=UUUDDUDDD bites=- face=root apexes=[d3,d4,d6,d7,d8] | 21 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | | C03 | word=UUDUUDDDD bites=- face=root apexes=[d2,d5,d6,d7,d8] | 25 | { `00012`, `00120`, `01112`, `01200`, `01222` } | -| C04 | word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] | 24 | { `00012`, `00102`, `00120`, `01002`, `01112`, `01200`, `01211`, `01222` } | -| C05 | word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] | 22 | { `00012`, `00102`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | -| C06 | word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] | 28 | { `00012`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01222` } | -| C07 | word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] | 27 | { `00012`, `00102`, `00120`, `01002`, `01112`, `01200`, `01211`, `01222` } | +| C04 | word=UUDUDUDDD bites=- face=root apexes=[d2,d4,d6,d7,d8] | 24 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | +| C05 | word=UUDUDDUDD bites=- face=root apexes=[d2,d4,d5,d7,d8] | 22 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | +| C06 | word=UUDUDDDUD bites=- face=root apexes=[d2,d4,d5,d6,d8] | 28 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | +| C07 | word=UUDDUUDDD bites=- face=root apexes=[d2,d3,d6,d7,d8] | 27 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | | C08 | word=UUDDUDUDD bites=- face=root apexes=[d2,d3,d5,d7,d8] | 18 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | | C09 | word=UDUDUDUDD bites=- face=root apexes=[d1,d3,d5,d7,d8] | 19 | { `00012`, `00102`, `00120`, `01002`, `01020`, `01112`, `01121`, `01200`, `01211`, `01222` } | diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.md index e8896de..2eba495 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.md @@ -1,10 +1,10 @@ # Inner-face down-apex sequence `000000` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 6×colour0. - Realised by **4** of 7 configs (M(T), inner face). -- **5** Kempe-balanced colourings (mod colour permutation) produce it. +- **5** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_000000.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. @@ -13,24 +13,24 @@ Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D 2 colouring(s) with down-apex sequence `000000`: -- face apexes (raw labels) `222222` → canonical `000000` · `A=012010101 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `111111` → canonical `000000` · `A=012020202 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `222222`, realises `000000` in some orientation · `A=012010101 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111111`, realises `000000` in some orientation · `A=012020202 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:1 d8:1]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] 1 colouring(s) with down-apex sequence `000000`: -- face apexes (raw labels) `111111` → canonical `000000` · `A=012020202 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111111`, realises `000000` in some orientation · `A=012020202 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:1 d8:1]` ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] 1 colouring(s) with down-apex sequence `000000`: -- face apexes (raw labels) `111111` → canonical `000000` · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111111`, realises `000000` in some orientation · `A=012020202 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:1 d8:1]` ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] 1 colouring(s) with down-apex sequence `000000`: -- face apexes (raw labels) `111111` → canonical `000000` · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111111`, realises `000000` in some orientation · `A=012020202 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.pdf index cccb951..ae2c626 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000000.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.md index bf89861..c244fff 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.md @@ -1,43 +1,88 @@ # Inner-face down-apex sequence `000011` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0, 2×colour1. -- Realised by **5** of 7 configs (M(T), inner face). -- **7** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **7** of 7 configs (M(T), inner face). +- **42** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_000011.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `000011`: +12 colouring(s) with down-apex sequence `000011`: -- face apexes (raw labels) `222211` → canonical `000011` · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `111122` → canonical `000011` · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `222211`, realises `000011` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `222002`, realises `000011` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `221122`, realises `000011` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `000011` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `200222`, realises `000011` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `000011` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `112222`, realises `000011` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `000011` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `000011` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `000011` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `000011` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `000011` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `000011`: +6 colouring(s) with down-apex sequence `000011`: -- face apexes (raw labels) `111122` → canonical `000011` · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112222`, realises `000011` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `000011` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `000011` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `000011` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `000011` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `000011` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `000011`: +6 colouring(s) with down-apex sequence `000011`: -- face apexes (raw labels) `111122` → canonical `000011` · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112222`, realises `000011` in some orientation · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `000011` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `000011` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `000011` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `002222`, realises `000011` in some orientation · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]` ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `000011`: +4 colouring(s) with down-apex sequence `000011`: -- face apexes (raw labels) `111122` → canonical `000011` · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `122111`, realises `000011` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `000011` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `000011` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +4 colouring(s) with down-apex sequence `000011`: + +- face apex colours (canonical-rep edge order) `200222`, realises `000011` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `000011` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `002222`, realises `000011` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +4 colouring(s) with down-apex sequence `000011`: + +- face apex colours (canonical-rep edge order) `211222`, realises `000011` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200222`, realises `000011` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `000011` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -2 colouring(s) with down-apex sequence `000011`: +6 colouring(s) with down-apex sequence `000011`: -- face apexes (raw labels) `000022` → canonical `000011` · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]` -- face apexes (raw labels) `000011` → canonical `000011` · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `221122`, realises `000011` in some orientation · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `000011` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `220022`, realises `000011` in some orientation · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `000011` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `000022`, realises `000011` in some orientation · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `000011`, realises `000011` in some orientation · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.pdf index bca85c0..49aec0f 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.png index f90542a..d2b983e 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000011.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.md index fa1ebdc..1680f5e 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.md @@ -1,17 +1,51 @@ # Inner-face down-apex sequence `000101` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0, 2×colour1. -- Realised by **1** of 7 configs (M(T), inner face). -- **1** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **4** of 7 configs (M(T), inner face). +- **20** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_000101.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `000101`: + +- face apex colours (canonical-rep edge order) `020222`, realises `000101` in some orientation · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `000101` in some orientation · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `000101`: + +- face apex colours (canonical-rep edge order) `020222`, realises `000101` in some orientation · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `000101` in some orientation · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `000101` in some orientation · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `000101` in some orientation · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `000101` in some orientation · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `000101` in some orientation · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +6 colouring(s) with down-apex sequence `000101`: + +- face apex colours (canonical-rep edge order) `212122`, realises `000101` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `000101` in some orientation · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `000101` in some orientation · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `000101` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `000101` in some orientation · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `000101` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]` + ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `000101`: +6 colouring(s) with down-apex sequence `000101`: -- face apexes (raw labels) `000202` → canonical `000101` · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `220202`, realises `000101` in some orientation · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `212122`, realises `000101` in some orientation · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `000101` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `011101`, realises `000101` in some orientation · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `000101` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `000202`, realises `000101` in some orientation · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.pdf index 2ea26ac..9c449ec 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.png index e4c988a..1ceab72 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000101.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.md index c2d3a49..f2b25f0 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.md @@ -1,36 +1,88 @@ # Inner-face down-apex sequence `000110` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0, 2×colour1. -- Realised by **4** of 7 configs (M(T), inner face). -- **5** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **7** of 7 configs (M(T), inner face). +- **42** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_000110.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `000110`: +12 colouring(s) with down-apex sequence `000110`: -- face apexes (raw labels) `222002` → canonical `000110` · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `111001` → canonical `000110` · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `222211`, realises `000110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `222002`, realises `000110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `221122`, realises `000110` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `000110` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `200222`, realises `000110` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `000110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `112222`, realises `000110` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `000110` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `000110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `000110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `000110` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `000110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `000110`: +6 colouring(s) with down-apex sequence `000110`: -- face apexes (raw labels) `111001` → canonical `000110` · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `112222`, realises `000110` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `000110` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `000110` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `000110` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `000110` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `000110` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `000110`: +6 colouring(s) with down-apex sequence `000110`: -- face apexes (raw labels) `111001` → canonical `000110` · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `112222`, realises `000110` in some orientation · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `000110` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `000110` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `000110` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `002222`, realises `000110` in some orientation · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]` ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `000110`: +4 colouring(s) with down-apex sequence `000110`: -- face apexes (raw labels) `111001` → canonical `000110` · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `122111`, realises `000110` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `000110` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `000110` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +4 colouring(s) with down-apex sequence `000110`: + +- face apex colours (canonical-rep edge order) `200222`, realises `000110` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `000110` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `002222`, realises `000110` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +4 colouring(s) with down-apex sequence `000110`: + +- face apex colours (canonical-rep edge order) `211222`, realises `000110` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200222`, realises `000110` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `000110` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]` + +## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] + +6 colouring(s) with down-apex sequence `000110`: + +- face apex colours (canonical-rep edge order) `221122`, realises `000110` in some orientation · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `000110` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `220022`, realises `000110` in some orientation · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `000110` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `000022`, realises `000110` in some orientation · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `000011`, realises `000110` in some orientation · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.pdf index e9a2772..4abfa82 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.png index 97eacab..ecabf70 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_000110.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.md index 2e2af0d..2d5c968 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.md @@ -1,10 +1,10 @@ # Inner-face down-apex sequence `001001` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0, 2×colour1. - Realised by **4** of 7 configs (M(T), inner face). -- **6** Kempe-balanced colourings (mod colour permutation) produce it. +- **8** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_001001.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. @@ -13,25 +13,27 @@ Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D 2 colouring(s) with down-apex sequence `001001`: -- face apexes (raw labels) `002002` → canonical `001001` · `A=012120121 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `001001` → canonical `001001` · `A=012120212 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `002002`, realises `001001` in some orientation · `A=012120121 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001001`, realises `001001` in some orientation · `A=012120212 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:0 d7:0 d8:1]` ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `001001`: +2 colouring(s) with down-apex sequence `001001`: -- face apexes (raw labels) `001001` → canonical `001001` · `A=012120212 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `122122`, realises `001001` in some orientation · `A=012010201 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001001`, realises `001001` in some orientation · `A=012120212 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:0 d7:0 d8:1]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] 2 colouring(s) with down-apex sequence `001001`: -- face apexes (raw labels) `002002` → canonical `001001` · `A=012120121 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `001001` → canonical `001001` · `A=012120212 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `002002`, realises `001001` in some orientation · `A=012120121 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001001`, realises `001001` in some orientation · `A=012120212 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:0 d7:0 d8:1]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `001001`: +2 colouring(s) with down-apex sequence `001001`: -- face apexes (raw labels) `001001` → canonical `001001` · `A=012120212 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `211211`, realises `001001` in some orientation · `A=010202102 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `001001`, realises `001001` in some orientation · `A=012120212 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:0 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.pdf index 9748fbe..e3b4f87 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.png index 58b64b1..7bc2431 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001001.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.md index b63c89b..c641479 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.md @@ -1,17 +1,51 @@ # Inner-face down-apex sequence `001010` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0, 2×colour1. -- Realised by **1** of 7 configs (M(T), inner face). -- **1** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **4** of 7 configs (M(T), inner face). +- **20** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_001010.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `001010`: + +- face apex colours (canonical-rep edge order) `020222`, realises `001010` in some orientation · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `001010` in some orientation · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `001010`: + +- face apex colours (canonical-rep edge order) `020222`, realises `001010` in some orientation · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `001010` in some orientation · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `001010` in some orientation · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `001010` in some orientation · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `001010` in some orientation · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `001010` in some orientation · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +6 colouring(s) with down-apex sequence `001010`: + +- face apex colours (canonical-rep edge order) `212122`, realises `001010` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `001010` in some orientation · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `001010` in some orientation · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `001010` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `001010` in some orientation · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `001010` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]` + ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `001010`: +6 colouring(s) with down-apex sequence `001010`: -- face apexes (raw labels) `220202` → canonical `001010` · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `220202`, realises `001010` in some orientation · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `212122`, realises `001010` in some orientation · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `001010` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `011101`, realises `001010` in some orientation · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `001010` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `000202`, realises `001010` in some orientation · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.pdf index 72b0e93..dbd1d62 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.png index 31f5ebd..80f00bd 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001010.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.md index 0f086ea..eb6a7a6 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.md @@ -1,37 +1,88 @@ # Inner-face down-apex sequence `001100` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0, 2×colour1. -- Realised by **4** of 7 configs (M(T), inner face). -- **6** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **7** of 7 configs (M(T), inner face). +- **42** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_001100.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `001100`: +12 colouring(s) with down-apex sequence `001100`: -- face apexes (raw labels) `221122` → canonical `001100` · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `112211` → canonical `001100` · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `222211`, realises `001100` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `222002`, realises `001100` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `221122`, realises `001100` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `001100` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `200222`, realises `001100` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `001100` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `112222`, realises `001100` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `001100` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `001100` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `001100` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `001100` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `001100` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `001100`: +6 colouring(s) with down-apex sequence `001100`: -- face apexes (raw labels) `112211` → canonical `001100` · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `112222`, realises `001100` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `001100` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `001100` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `001100` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `001100` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `001100` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `001100`: +6 colouring(s) with down-apex sequence `001100`: -- face apexes (raw labels) `112211` → canonical `001100` · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `112222`, realises `001100` in some orientation · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `001100` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `001100` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `001100` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `002222`, realises `001100` in some orientation · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `001100` in some orientation · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]` + +## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] + +4 colouring(s) with down-apex sequence `001100`: + +- face apex colours (canonical-rep edge order) `122111`, realises `001100` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `001100` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `001100` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `001111`, realises `001100` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +4 colouring(s) with down-apex sequence `001100`: + +- face apex colours (canonical-rep edge order) `200222`, realises `001100` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `001100` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `002222`, realises `001100` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `001100` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +4 colouring(s) with down-apex sequence `001100`: + +- face apex colours (canonical-rep edge order) `211222`, realises `001100` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200222`, realises `001100` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `001100` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `001100` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -2 colouring(s) with down-apex sequence `001100`: +6 colouring(s) with down-apex sequence `001100`: -- face apexes (raw labels) `221122` → canonical `001100` · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]` -- face apexes (raw labels) `220022` → canonical `001100` · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221122`, realises `001100` in some orientation · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `001100` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `220022`, realises `001100` in some orientation · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `001100` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `000022`, realises `001100` in some orientation · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `000011`, realises `001100` in some orientation · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.pdf index 41ac6f9..dcdd4e5 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.png index cfa8928..fa35a0c 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001100.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.md index 67247d4..4e5877b 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.md @@ -1,58 +1,88 @@ # Inner-face down-apex sequence `001111` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 4×colour1. - Realised by **7** of 7 configs (M(T), inner face). -- **12** Kempe-balanced colourings (mod colour permutation) produce it. +- **42** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_001111.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `001111`: +12 colouring(s) with down-apex sequence `001111`: -- face apexes (raw labels) `221111` → canonical `001111` · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `112222` → canonical `001111` · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `222211`, realises `001111` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `222002`, realises `001111` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `221122`, realises `001111` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `001111` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `200222`, realises `001111` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `001111` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `112222`, realises `001111` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `001111` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `001111` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `001111` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `001111` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `001111` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `001111`: +6 colouring(s) with down-apex sequence `001111`: -- face apexes (raw labels) `112222` → canonical `001111` · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112222`, realises `001111` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `001111` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `001111` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `001111` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `001111` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `001111` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] -3 colouring(s) with down-apex sequence `001111`: +6 colouring(s) with down-apex sequence `001111`: -- face apexes (raw labels) `112222` → canonical `001111` · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `002222` → canonical `001111` · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `112222`, realises `001111` in some orientation · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `001111` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `001111` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `001111` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `002222`, realises `001111` in some orientation · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `001111` in some orientation · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]` ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `001111`: +4 colouring(s) with down-apex sequence `001111`: -- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `122111`, realises `001111` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `001111` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `001111` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `001111`, realises `001111` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `001111`: +4 colouring(s) with down-apex sequence `001111`: -- face apexes (raw labels) `002222` → canonical `001111` · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `200222`, realises `001111` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `001111` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `002222`, realises `001111` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `001111` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `001111`: +4 colouring(s) with down-apex sequence `001111`: -- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `211222`, realises `001111` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200222`, realises `001111` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `001111` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `001111` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -2 colouring(s) with down-apex sequence `001111`: +6 colouring(s) with down-apex sequence `001111`: -- face apexes (raw labels) `221111` → canonical `001111` · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]` -- face apexes (raw labels) `001111` → canonical `001111` · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `221122`, realises `001111` in some orientation · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `001111` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `220022`, realises `001111` in some orientation · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `001111` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `000022`, realises `001111` in some orientation · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `000011`, realises `001111` in some orientation · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.pdf index 621185c..db82438 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.png index 044616b..0cba09b 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001111.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.md index b066d1e..23318ea 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.md @@ -1,10 +1,10 @@ # Inner-face down-apex sequence `001122` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. - Realised by **5** of 7 configs (M(T), inner face). -- **8** Kempe-balanced colourings (mod colour permutation) produce it. +- **11** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_001122.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. @@ -13,32 +13,35 @@ Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D 2 colouring(s) with down-apex sequence `001122`: -- face apexes (raw labels) `002211` → canonical `001122` · `A=012120102 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `002211`, realises `001122` in some orientation · `A=012120102 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `001122`, realises `001122` in some orientation · `A=012120201 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:2 d8:2]` ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `001122`: +2 colouring(s) with down-apex sequence `001122`: -- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `122001`, realises `001122` in some orientation · `A=012010212 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `001122`, realises `001122` in some orientation · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:2 d8:2]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] 2 colouring(s) with down-apex sequence `001122`: -- face apexes (raw labels) `002211` → canonical `001122` · `A=012120102 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `002211`, realises `001122` in some orientation · `A=012120102 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `001122`, realises `001122` in some orientation · `A=012120201 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:2 d8:2]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `001122`: +2 colouring(s) with down-apex sequence `001122`: -- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `211002`, realises `001122` in some orientation · `A=010202121 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001122`, realises `001122` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:2 d8:2]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -2 colouring(s) with down-apex sequence `001122`: +3 colouring(s) with down-apex sequence `001122`: -- face apexes (raw labels) `220011` → canonical `001122` · `A=010121202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:1 d8:1]` -- face apexes (raw labels) `001122` → canonical `001122` · `A=012120201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `220011`, realises `001122` in some orientation · `A=010121202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `211002`, realises `001122` in some orientation · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001122`, realises `001122` in some orientation · `A=012120201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:2 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.pdf index fb4b2c8..4a55cea 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.png index 71a93b2..a37f489 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001122.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001212.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001212.md new file mode 100644 index 0000000..0dbc75b --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001212.md @@ -0,0 +1,53 @@ +# Inner-face down-apex sequence `001212` + +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. + +- Colour multiset: 2×colour0, 2×colour1, 2×colour2. +- Realised by **5** of 7 configs (M(T), inner face). +- **17** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. +- Figure: `seq_001212.png` (black rings mark the face's down apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. + +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `001212`: + +- face apex colours (canonical-rep edge order) `021102`, realises `001212` in some orientation · `A=012102021 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `001212` in some orientation · `A=012102102 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]` + +## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] + +2 colouring(s) with down-apex sequence `001212`: + +- face apex colours (canonical-rep edge order) `120201`, realises `001212` in some orientation · `A=012012012 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `021102`, realises `001212` in some orientation · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:1 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `001212`: + +- face apex colours (canonical-rep edge order) `021102`, realises `001212` in some orientation · `A=012012021 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `001212` in some orientation · `A=012012102 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `012201`, realises `001212` in some orientation · `A=012021012 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `001212` in some orientation · `A=012021201 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `001212` in some orientation · `A=012102021 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `001212` in some orientation · `A=012102102 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +4 colouring(s) with down-apex sequence `001212`: + +- face apex colours (canonical-rep edge order) `212001`, realises `001212` in some orientation · `A=010201212 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020211`, realises `001212` in some orientation · `A=012012102 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `001212` in some orientation · `A=012021201 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `001212` in some orientation · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:1 d7:0 d8:2]` + +## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] + +3 colouring(s) with down-apex sequence `001212`: + +- face apex colours (canonical-rep edge order) `212001`, realises `001212` in some orientation · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `210102`, realises `001212` in some orientation · `A=010212021 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:0 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `011202`, realises `001212` in some orientation · `A=012020121 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:2 d7:0 d8:2]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001212.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001212.pdf new file mode 100644 index 0000000..dc7dd14 Binary files /dev/null and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001212.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001212.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001212.png new file mode 100644 index 0000000..b812e7e Binary files /dev/null and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001212.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.md index 2ff418e..162ea27 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.md @@ -1,30 +1,49 @@ # Inner-face down-apex sequence `001221` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **3** of 7 configs (M(T), inner face). -- **4** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **5** of 7 configs (M(T), inner face). +- **13** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_001221.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `001221`: +6 colouring(s) with down-apex sequence `001221`: -- face apexes (raw labels) `221001` → canonical `001221` · `A=012010212 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `112002` → canonical `001221` · `A=012020121 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `221001`, realises `001221` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201102`, realises `001221` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `200211`, realises `001221` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `112002`, realises `001221` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102201`, realises `001221` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100122`, realises `001221` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:2 d8:2]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `001221`: +3 colouring(s) with down-apex sequence `001221`: -- face apexes (raw labels) `112002` → canonical `001221` · `A=012020121 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `112002`, realises `001221` in some orientation · `A=012020121 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102201`, realises `001221` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100122`, realises `001221` in some orientation · `A=012021201 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:2 d8:2]` ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] 1 colouring(s) with down-apex sequence `001221`: -- face apexes (raw labels) `112002` → canonical `001221` · `A=012020121 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `112002`, realises `001221` in some orientation · `A=012020121 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:0 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `001221`: + +- face apex colours (canonical-rep edge order) `201102`, realises `001221` in some orientation · `A=010212021 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `200211`, realises `001221` in some orientation · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:1 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +1 colouring(s) with down-apex sequence `001221`: + +- face apex colours (canonical-rep edge order) `200211`, realises `001221` in some orientation · `A=010212102 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.pdf index d02e819..41d5679 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.png index ab9ab18..f53f7dd 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_001221.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.md index 2b6b022..17068f4 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.md @@ -1,38 +1,51 @@ # Inner-face down-apex sequence `010001` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0, 2×colour1. - Realised by **4** of 7 configs (M(T), inner face). -- **7** Kempe-balanced colourings (mod colour permutation) produce it. +- **20** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_010001.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `010001`: +2 colouring(s) with down-apex sequence `010001`: -- face apexes (raw labels) `020002` → canonical `010001` · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020222`, realises `010001` in some orientation · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010001` in some orientation · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] -3 colouring(s) with down-apex sequence `010001`: +6 colouring(s) with down-apex sequence `010001`: -- face apexes (raw labels) `020002` → canonical `010001` · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `010001` → canonical `010001` · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]` -- face apexes (raw labels) `020002` → canonical `010001` · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020222`, realises `010001` in some orientation · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010001` in some orientation · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `010001` in some orientation · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `010001` in some orientation · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `010001` in some orientation · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010001` in some orientation · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -2 colouring(s) with down-apex sequence `010001`: +6 colouring(s) with down-apex sequence `010001`: -- face apexes (raw labels) `020002` → canonical `010001` · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `010001` → canonical `010001` · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `212122`, realises `010001` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `010001` in some orientation · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `010001` in some orientation · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010001` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `010001` in some orientation · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `010001` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `010001`: +6 colouring(s) with down-apex sequence `010001`: -- face apexes (raw labels) `010001` → canonical `010001` · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `220202`, realises `010001` in some orientation · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `212122`, realises `010001` in some orientation · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `010001` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `011101`, realises `010001` in some orientation · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `010001` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `000202`, realises `010001` in some orientation · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.pdf index c602620..4fa979c 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.png index b7e4022..7d38a74 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010001.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010010.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010010.md new file mode 100644 index 0000000..8225c9f --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010010.md @@ -0,0 +1,39 @@ +# Inner-face down-apex sequence `010010` + +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. + +- Colour multiset: 4×colour0, 2×colour1. +- Realised by **4** of 7 configs (M(T), inner face). +- **8** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. +- Figure: `seq_010010.png` (black rings mark the face's down apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. + +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `010010`: + +- face apex colours (canonical-rep edge order) `002002`, realises `010010` in some orientation · `A=012120121 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001001`, realises `010010` in some orientation · `A=012120212 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:0 d7:0 d8:1]` + +## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] + +2 colouring(s) with down-apex sequence `010010`: + +- face apex colours (canonical-rep edge order) `122122`, realises `010010` in some orientation · `A=012010201 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001001`, realises `010010` in some orientation · `A=012120212 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:0 d7:0 d8:1]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `010010`: + +- face apex colours (canonical-rep edge order) `002002`, realises `010010` in some orientation · `A=012120121 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001001`, realises `010010` in some orientation · `A=012120212 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:0 d7:0 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +2 colouring(s) with down-apex sequence `010010`: + +- face apex colours (canonical-rep edge order) `211211`, realises `010010` in some orientation · `A=010202102 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `001001`, realises `010010` in some orientation · `A=012120212 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:0 d7:0 d8:1]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010010.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010010.pdf new file mode 100644 index 0000000..b90b3f9 Binary files /dev/null and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010010.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010010.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010010.png new file mode 100644 index 0000000..3d3cba0 Binary files /dev/null and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010010.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.md index fdf29fe..902d2c3 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.md @@ -1,23 +1,51 @@ # Inner-face down-apex sequence `010100` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0, 2×colour1. -- Realised by **2** of 7 configs (M(T), inner face). -- **2** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **4** of 7 configs (M(T), inner face). +- **20** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_010100.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `010100`: + +- face apex colours (canonical-rep edge order) `020222`, realises `010100` in some orientation · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010100` in some orientation · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `010100`: + +- face apex colours (canonical-rep edge order) `020222`, realises `010100` in some orientation · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010100` in some orientation · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `010100` in some orientation · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `010100` in some orientation · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `010100` in some orientation · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010100` in some orientation · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` + ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `010100`: +6 colouring(s) with down-apex sequence `010100`: -- face apexes (raw labels) `212122` → canonical `010100` · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212122`, realises `010100` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `010100` in some orientation · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `010100` in some orientation · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010100` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `010100` in some orientation · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `010100` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `010100`: +6 colouring(s) with down-apex sequence `010100`: -- face apexes (raw labels) `212122` → canonical `010100` · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `220202`, realises `010100` in some orientation · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `212122`, realises `010100` in some orientation · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `010100` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `011101`, realises `010100` in some orientation · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `010100` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `000202`, realises `010100` in some orientation · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.pdf index 3de96fb..d428910 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.png index da59dfb..ef87beb 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010100.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.md index 64d447c..da6143d 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.md @@ -1,39 +1,51 @@ # Inner-face down-apex sequence `010111` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 4×colour1. - Realised by **4** of 7 configs (M(T), inner face). -- **8** Kempe-balanced colourings (mod colour permutation) produce it. +- **20** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_010111.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `010111`: +2 colouring(s) with down-apex sequence `010111`: -- face apexes (raw labels) `020222` → canonical `010111` · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020222`, realises `010111` in some orientation · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010111` in some orientation · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] -3 colouring(s) with down-apex sequence `010111`: +6 colouring(s) with down-apex sequence `010111`: -- face apexes (raw labels) `020222` → canonical `010111` · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `010111` → canonical `010111` · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `020222` → canonical `010111` · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020222`, realises `010111` in some orientation · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010111` in some orientation · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `010111` in some orientation · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `010111` in some orientation · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `010111` in some orientation · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010111` in some orientation · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -3 colouring(s) with down-apex sequence `010111`: +6 colouring(s) with down-apex sequence `010111`: -- face apexes (raw labels) `212111` → canonical `010111` · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]` -- face apexes (raw labels) `020222` → canonical `010111` · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `010111` → canonical `010111` · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `212122`, realises `010111` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `010111` in some orientation · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `010111` in some orientation · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `010111` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `010111` in some orientation · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `010111` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `010111`: +6 colouring(s) with down-apex sequence `010111`: -- face apexes (raw labels) `212111` → canonical `010111` · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `220202`, realises `010111` in some orientation · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `212122`, realises `010111` in some orientation · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `010111` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `011101`, realises `010111` in some orientation · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `010111` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `000202`, realises `010111` in some orientation · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.pdf index 53b7f47..c77ba9f 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.png index 5d884ff..949ffaf 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010111.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.md index 92c6b47..85257c4 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.md @@ -1,32 +1,53 @@ # Inner-face down-apex sequence `010122` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **3** of 7 configs (M(T), inner face). -- **6** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **5** of 7 configs (M(T), inner face). +- **17** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_010122.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `010122`: +2 colouring(s) with down-apex sequence `010122`: -- face apexes (raw labels) `020211` → canonical `010122` · `A=012102102 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `021102`, realises `010122` in some orientation · `A=012102021 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `010122` in some orientation · `A=012102102 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]` -## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] - -3 colouring(s) with down-apex sequence `010122`: - -- face apexes (raw labels) `020211` → canonical `010122` · `A=012012102 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `010122` → canonical `010122` · `A=012021201 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]` -- face apexes (raw labels) `020211` → canonical `010122` · `A=012102102 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` - -## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] +## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] 2 colouring(s) with down-apex sequence `010122`: -- face apexes (raw labels) `020211` → canonical `010122` · `A=012012102 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `010122` → canonical `010122` · `A=012021201 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `120201`, realises `010122` in some orientation · `A=012012012 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `021102`, realises `010122` in some orientation · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:1 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `010122`: + +- face apex colours (canonical-rep edge order) `021102`, realises `010122` in some orientation · `A=012012021 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `010122` in some orientation · `A=012012102 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `012201`, realises `010122` in some orientation · `A=012021012 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `010122` in some orientation · `A=012021201 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `010122` in some orientation · `A=012102021 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `010122` in some orientation · `A=012102102 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +4 colouring(s) with down-apex sequence `010122`: + +- face apex colours (canonical-rep edge order) `212001`, realises `010122` in some orientation · `A=010201212 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020211`, realises `010122` in some orientation · `A=012012102 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `010122` in some orientation · `A=012021201 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `010122` in some orientation · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:1 d7:0 d8:2]` + +## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] + +3 colouring(s) with down-apex sequence `010122`: + +- face apex colours (canonical-rep edge order) `212001`, realises `010122` in some orientation · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `210102`, realises `010122` in some orientation · `A=010212021 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:0 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `011202`, realises `010122` in some orientation · `A=012020121 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.pdf index d54aa82..64849cd 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.png index 28b7c11..a4e0908 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010122.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010212.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010212.md new file mode 100644 index 0000000..a3c32bc --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010212.md @@ -0,0 +1,38 @@ +# Inner-face down-apex sequence `010212` + +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. + +- Colour multiset: 2×colour0, 2×colour1, 2×colour2. +- Realised by **4** of 7 configs (M(T), inner face). +- **7** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. +- Figure: `seq_010212.png` (black rings mark the face's down apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. + +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +1 colouring(s) with down-apex sequence `010212`: + +- face apex colours (canonical-rep edge order) `021201`, realises `010212` in some orientation · `A=012102012 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:2 d7:0 d8:1]` + +## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] + +2 colouring(s) with down-apex sequence `010212`: + +- face apex colours (canonical-rep edge order) `120102`, realises `010212` in some orientation · `A=012012021 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `021201`, realises `010212` in some orientation · `A=012102012 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:2 d7:0 d8:1]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +3 colouring(s) with down-apex sequence `010212`: + +- face apex colours (canonical-rep edge order) `021201`, realises `010212` in some orientation · `A=012012012 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `012102`, realises `010212` in some orientation · `A=012021021 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `021201`, realises `010212` in some orientation · `A=012102012 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +1 colouring(s) with down-apex sequence `010212`: + +- face apex colours (canonical-rep edge order) `021201`, realises `010212` in some orientation · `A=012102012 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:2 d7:0 d8:1]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010212.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010212.pdf new file mode 100644 index 0000000..38b5186 Binary files /dev/null and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010212.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010212.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010212.png new file mode 100644 index 0000000..c3ac9b1 Binary files /dev/null and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010212.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.md index 4401afc..caff51e 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.md @@ -1,23 +1,53 @@ # Inner-face down-apex sequence `010221` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **2** of 7 configs (M(T), inner face). -- **2** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **5** of 7 configs (M(T), inner face). +- **17** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_010221.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `010221`: + +- face apex colours (canonical-rep edge order) `021102`, realises `010221` in some orientation · `A=012102021 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `010221` in some orientation · `A=012102102 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]` + +## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] + +2 colouring(s) with down-apex sequence `010221`: + +- face apex colours (canonical-rep edge order) `120201`, realises `010221` in some orientation · `A=012012012 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `021102`, realises `010221` in some orientation · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:1 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `010221`: + +- face apex colours (canonical-rep edge order) `021102`, realises `010221` in some orientation · `A=012012021 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `010221` in some orientation · `A=012012102 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `012201`, realises `010221` in some orientation · `A=012021012 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `010221` in some orientation · `A=012021201 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `010221` in some orientation · `A=012102021 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `010221` in some orientation · `A=012102102 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` + ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `010221`: +4 colouring(s) with down-apex sequence `010221`: -- face apexes (raw labels) `212001` → canonical `010221` · `A=010201212 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `212001`, realises `010221` in some orientation · `A=010201212 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020211`, realises `010221` in some orientation · `A=012012102 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `010221` in some orientation · `A=012021201 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `010221` in some orientation · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:1 d7:0 d8:2]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `010221`: +3 colouring(s) with down-apex sequence `010221`: -- face apexes (raw labels) `212001` → canonical `010221` · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `212001`, realises `010221` in some orientation · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `210102`, realises `010221` in some orientation · `A=010212021 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:0 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `011202`, realises `010221` in some orientation · `A=012020121 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.pdf index efc69fe..1997f96 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.png index e5cbc4d..fcdcd05 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_010221.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.md index ce7099b..a1fc350 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.md @@ -1,43 +1,88 @@ # Inner-face down-apex sequence `011000` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 4×colour0, 2×colour1. -- Realised by **5** of 7 configs (M(T), inner face). -- **7** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **7** of 7 configs (M(T), inner face). +- **42** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_011000.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `011000`: +12 colouring(s) with down-apex sequence `011000`: -- face apexes (raw labels) `200222` → canonical `011000` · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `100111` → canonical `011000` · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `222211`, realises `011000` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `222002`, realises `011000` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `221122`, realises `011000` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `011000` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `200222`, realises `011000` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `011000` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `112222`, realises `011000` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `011000` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `011000` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `011000` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `011000` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `011000` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `011000`: +6 colouring(s) with down-apex sequence `011000`: -- face apexes (raw labels) `100111` → canonical `011000` · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `112222`, realises `011000` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `011000` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `011000` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `011000` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `011000` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `011000` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]` + +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `011000`: + +- face apex colours (canonical-rep edge order) `112222`, realises `011000` in some orientation · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `011000` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `011000` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `011000` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `002222`, realises `011000` in some orientation · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `011000` in some orientation · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]` ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `011000`: +4 colouring(s) with down-apex sequence `011000`: -- face apexes (raw labels) `122111` → canonical `011000` · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `122111`, realises `011000` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `011000` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `011000` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `001111`, realises `011000` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `011000`: +4 colouring(s) with down-apex sequence `011000`: -- face apexes (raw labels) `200222` → canonical `011000` · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200222`, realises `011000` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `011000` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `002222`, realises `011000` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `011000` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -2 colouring(s) with down-apex sequence `011000`: +4 colouring(s) with down-apex sequence `011000`: -- face apexes (raw labels) `211222` → canonical `011000` · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]` -- face apexes (raw labels) `200222` → canonical `011000` · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `211222`, realises `011000` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200222`, realises `011000` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `011000` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `011000` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]` + +## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] + +6 colouring(s) with down-apex sequence `011000`: + +- face apex colours (canonical-rep edge order) `221122`, realises `011000` in some orientation · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `011000` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `220022`, realises `011000` in some orientation · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `011000` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `000022`, realises `011000` in some orientation · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `000011`, realises `011000` in some orientation · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.pdf index f25d406..bceb245 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.png index 96a8959..90ed0c8 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011000.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.md index 95a2e5c..9734d9e 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.md @@ -1,23 +1,39 @@ # Inner-face down-apex sequence `011011` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 4×colour1. -- Realised by **2** of 7 configs (M(T), inner face). -- **2** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **4** of 7 configs (M(T), inner face). +- **8** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_011011.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `011011`: + +- face apex colours (canonical-rep edge order) `002002`, realises `011011` in some orientation · `A=012120121 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001001`, realises `011011` in some orientation · `A=012120212 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:0 d7:0 d8:1]` + ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `011011`: +2 colouring(s) with down-apex sequence `011011`: -- face apexes (raw labels) `122122` → canonical `011011` · `A=012010201 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `122122`, realises `011011` in some orientation · `A=012010201 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001001`, realises `011011` in some orientation · `A=012120212 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:0 d7:0 d8:1]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `011011`: + +- face apex colours (canonical-rep edge order) `002002`, realises `011011` in some orientation · `A=012120121 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001001`, realises `011011` in some orientation · `A=012120212 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:0 d7:0 d8:1]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `011011`: +2 colouring(s) with down-apex sequence `011011`: -- face apexes (raw labels) `211211` → canonical `011011` · `A=010202102 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `211211`, realises `011011` in some orientation · `A=010202102 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `001001`, realises `011011` in some orientation · `A=012120212 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:0 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.pdf index f136350..c994de5 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.png index b4761b2..37e6af8 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011011.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.md index faa5dd9..65d0032 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.md @@ -1,36 +1,49 @@ # Inner-face down-apex sequence `011022` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **4** of 7 configs (M(T), inner face). -- **5** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **5** of 7 configs (M(T), inner face). +- **13** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_011022.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `011022`: +6 colouring(s) with down-apex sequence `011022`: -- face apexes (raw labels) `200211` → canonical `011022` · `A=012012102 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:1 d8:1]` -- face apexes (raw labels) `100122` → canonical `011022` · `A=012021201 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221001`, realises `011022` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201102`, realises `011022` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `200211`, realises `011022` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `112002`, realises `011022` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102201`, realises `011022` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100122`, realises `011022` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:2 d8:2]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] +3 colouring(s) with down-apex sequence `011022`: + +- face apex colours (canonical-rep edge order) `112002`, realises `011022` in some orientation · `A=012020121 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102201`, realises `011022` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100122`, realises `011022` in some orientation · `A=012021201 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:2 d8:2]` + +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + 1 colouring(s) with down-apex sequence `011022`: -- face apexes (raw labels) `100122` → canonical `011022` · `A=012021201 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112002`, realises `011022` in some orientation · `A=012020121 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:0 d7:0 d8:2]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `011022`: +2 colouring(s) with down-apex sequence `011022`: -- face apexes (raw labels) `200211` → canonical `011022` · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `201102`, realises `011022` in some orientation · `A=010212021 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `200211`, realises `011022` in some orientation · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:1 d8:1]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] 1 colouring(s) with down-apex sequence `011022`: -- face apexes (raw labels) `200211` → canonical `011022` · `A=010212102 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `200211`, realises `011022` in some orientation · `A=010212102 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.pdf index 980f75f..1914dab 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.png index 1fdc3f7..95c03e2 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011022.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.md index fa6d51d..884892a 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.md @@ -1,17 +1,51 @@ # Inner-face down-apex sequence `011101` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 4×colour1. -- Realised by **1** of 7 configs (M(T), inner face). -- **1** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **4** of 7 configs (M(T), inner face). +- **20** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_011101.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `011101`: + +- face apex colours (canonical-rep edge order) `020222`, realises `011101` in some orientation · `A=012102101 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `011101` in some orientation · `A=012102121 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:0 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `011101`: + +- face apex colours (canonical-rep edge order) `020222`, realises `011101` in some orientation · `A=012012101 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `011101` in some orientation · `A=012012121 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `011101` in some orientation · `A=012021202 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `011101` in some orientation · `A=012021212 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `011101` in some orientation · `A=012102101 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `011101` in some orientation · `A=012102121 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:0 d7:0 d8:2]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +6 colouring(s) with down-apex sequence `011101`: + +- face apex colours (canonical-rep edge order) `212122`, realises `011101` in some orientation · `A=010201201 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `011101` in some orientation · `A=010201202 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `020222`, realises `011101` in some orientation · `A=012012101 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `020002`, realises `011101` in some orientation · `A=012012121 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `010111`, realises `011101` in some orientation · `A=012021202 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `011101` in some orientation · `A=012021212 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:0 d7:0 d8:1]` + ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `011101`: +6 colouring(s) with down-apex sequence `011101`: -- face apexes (raw labels) `011101` → canonical `011101` · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `220202`, realises `011101` in some orientation · `A=010121021 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `212122`, realises `011101` in some orientation · `A=010210201 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `212111`, realises `011101` in some orientation · `A=010210202 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:2 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `011101`, realises `011101` in some orientation · `A=012020212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:1 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010001`, realises `011101` in some orientation · `A=012021212 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:0 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `000202`, realises `011101` in some orientation · `A=012121021 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.pdf index e692b2b..15ee0f5 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.png index a53c534..9062218 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011101.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.md index 099c7c1..b64cc2e 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.md @@ -1,36 +1,88 @@ # Inner-face down-apex sequence `011110` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 4×colour1. -- Realised by **4** of 7 configs (M(T), inner face). -- **5** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **7** of 7 configs (M(T), inner face). +- **42** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_011110.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `011110`: +12 colouring(s) with down-apex sequence `011110`: -- face apexes (raw labels) `200002` → canonical `011110` · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]` -- face apexes (raw labels) `100001` → canonical `011110` · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `222211`, realises `011110` in some orientation · `A=012010102 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `222002`, realises `011110` in some orientation · `A=012010121 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `221122`, realises `011110` in some orientation · `A=012010201 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `011110` in some orientation · `A=012010202 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `200222`, realises `011110` in some orientation · `A=012012101 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `011110` in some orientation · `A=012012121 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `112222`, realises `011110` in some orientation · `A=012020101 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `011110` in some orientation · `A=012020102 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `011110` in some orientation · `A=012020201 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `011110` in some orientation · `A=012020212 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `011110` in some orientation · `A=012021202 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `011110` in some orientation · `A=012021212 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:0 d7:0 d8:1]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `011110`: +6 colouring(s) with down-apex sequence `011110`: -- face apexes (raw labels) `100001` → canonical `011110` · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `112222`, realises `011110` in some orientation · `A=012020101 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `011110` in some orientation · `A=012020102 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `011110` in some orientation · `A=012020201 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `011110` in some orientation · `A=012020212 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100111`, realises `011110` in some orientation · `A=012021202 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `100001`, realises `011110` in some orientation · `A=012021212 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:0 d7:0 d8:1]` + +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `011110`: + +- face apex colours (canonical-rep edge order) `112222`, realises `011110` in some orientation · `A=012020101 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `112211`, realises `011110` in some orientation · `A=012020102 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `011110` in some orientation · `A=012020201 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `011110` in some orientation · `A=012020212 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `002222`, realises `011110` in some orientation · `A=012120101 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `011110` in some orientation · `A=012120202 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:1 d8:1]` + +## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] + +4 colouring(s) with down-apex sequence `011110`: + +- face apex colours (canonical-rep edge order) `122111`, realises `011110` in some orientation · `A=012010202 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `111122`, realises `011110` in some orientation · `A=012020201 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `111001`, realises `011110` in some orientation · `A=012020212 U[u0:2 u1:0 u5:1] D[d2:1 d3:1 d4:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `001111`, realises `011110` in some orientation · `A=012120202 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:1 d8:1]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `011110`: +4 colouring(s) with down-apex sequence `011110`: -- face apexes (raw labels) `200002` → canonical `011110` · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `200222`, realises `011110` in some orientation · `A=010212101 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `011110` in some orientation · `A=010212121 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `002222`, realises `011110` in some orientation · `A=012120101 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `011110` in some orientation · `A=012120202 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:1 d8:1]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `011110`: +4 colouring(s) with down-apex sequence `011110`: -- face apexes (raw labels) `200002` → canonical `011110` · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `211222`, realises `011110` in some orientation · `A=010202101 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200222`, realises `011110` in some orientation · `A=010212101 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `200002`, realises `011110` in some orientation · `A=010212121 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `011110` in some orientation · `A=012120202 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:1 d8:1]` + +## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] + +6 colouring(s) with down-apex sequence `011110`: + +- face apex colours (canonical-rep edge order) `221122`, realises `011110` in some orientation · `A=010120201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `221111`, realises `011110` in some orientation · `A=010120202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `220022`, realises `011110` in some orientation · `A=010121201 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `001111`, realises `011110` in some orientation · `A=012120202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `000022`, realises `011110` in some orientation · `A=012121201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `000011`, realises `011110` in some orientation · `A=012121202 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:0 d5:0 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.pdf index 145bc57..998aa58 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.png index ebbfa3d..5597a03 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011110.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.md index 173a3d6..48856b6 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.md @@ -1,17 +1,53 @@ # Inner-face down-apex sequence `011202` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **1** of 7 configs (M(T), inner face). -- **1** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **5** of 7 configs (M(T), inner face). +- **17** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_011202.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `011202`: + +- face apex colours (canonical-rep edge order) `021102`, realises `011202` in some orientation · `A=012102021 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `011202` in some orientation · `A=012102102 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]` + +## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] + +2 colouring(s) with down-apex sequence `011202`: + +- face apex colours (canonical-rep edge order) `120201`, realises `011202` in some orientation · `A=012012012 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `021102`, realises `011202` in some orientation · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:1 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `011202`: + +- face apex colours (canonical-rep edge order) `021102`, realises `011202` in some orientation · `A=012012021 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `011202` in some orientation · `A=012012102 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `012201`, realises `011202` in some orientation · `A=012021012 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `011202` in some orientation · `A=012021201 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `011202` in some orientation · `A=012102021 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `011202` in some orientation · `A=012102102 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +4 colouring(s) with down-apex sequence `011202`: + +- face apex colours (canonical-rep edge order) `212001`, realises `011202` in some orientation · `A=010201212 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020211`, realises `011202` in some orientation · `A=012012102 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `011202` in some orientation · `A=012021201 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `011202` in some orientation · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:1 d7:0 d8:2]` + ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `011202`: +3 colouring(s) with down-apex sequence `011202`: -- face apexes (raw labels) `011202` → canonical `011202` · `A=012020121 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:2 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `212001`, realises `011202` in some orientation · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `210102`, realises `011202` in some orientation · `A=010212021 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:0 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `011202`, realises `011202` in some orientation · `A=012020121 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.pdf index a709db6..c91dcf8 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.png index add4999..0ac6412 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011202.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.md index c7443fc..edf3a38 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.md @@ -1,29 +1,47 @@ # Inner-face down-apex sequence `011220` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **3** of 7 configs (M(T), inner face). -- **3** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **5** of 7 configs (M(T), inner face). +- **11** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_011220.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `011220`: + +- face apex colours (canonical-rep edge order) `002211`, realises `011220` in some orientation · `A=012120102 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `001122`, realises `011220` in some orientation · `A=012120201 U[u0:2 u1:0 u4:1] D[d2:0 d3:0 d5:1 d6:1 d7:2 d8:2]` + ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `011220`: +2 colouring(s) with down-apex sequence `011220`: -- face apexes (raw labels) `122001` → canonical `011220` · `A=012010212 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `122001`, realises `011220` in some orientation · `A=012010212 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `001122`, realises `011220` in some orientation · `A=012120201 U[u0:2 u1:0 u5:1] D[d2:0 d3:0 d4:1 d6:1 d7:2 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `011220`: + +- face apex colours (canonical-rep edge order) `002211`, realises `011220` in some orientation · `A=012120102 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:2 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `001122`, realises `011220` in some orientation · `A=012120201 U[u0:2 u2:0 u4:1] D[d1:0 d3:0 d5:1 d6:1 d7:2 d8:2]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `011220`: +2 colouring(s) with down-apex sequence `011220`: -- face apexes (raw labels) `211002` → canonical `011220` · `A=010202121 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `211002`, realises `011220` in some orientation · `A=010202121 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:1 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001122`, realises `011220` in some orientation · `A=012120201 U[u0:2 u2:0 u5:1] D[d1:0 d3:0 d4:1 d6:1 d7:2 d8:2]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `011220`: +3 colouring(s) with down-apex sequence `011220`: -- face apexes (raw labels) `211002` → canonical `011220` · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `220011`, realises `011220` in some orientation · `A=010121202 U[u0:2 u3:0 u6:1] D[d1:2 d2:2 d4:0 d5:0 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `211002`, realises `011220` in some orientation · `A=010202121 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:1 d5:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `001122`, realises `011220` in some orientation · `A=012120201 U[u0:2 u3:0 u6:1] D[d1:0 d2:0 d4:1 d5:1 d7:2 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.pdf index 8eb2d0b..55afa42 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.png index 7411a47..6f8e4b8 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_011220.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.md index e61c77c..743ef47 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.md @@ -1,10 +1,10 @@ # Inner-face down-apex sequence `012012` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. - Realised by **3** of 7 configs (M(T), inner face). -- **4** Kempe-balanced colourings (mod colour permutation) produce it. +- **4** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_012012.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. @@ -13,18 +13,18 @@ Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D 2 colouring(s) with down-apex sequence `012012`: -- face apexes (raw labels) `201201` → canonical `012012` · `A=012012012 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `102102` → canonical `012012` · `A=012021021 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `201201`, realises `012012` in some orientation · `A=012012012 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `102102`, realises `012012` in some orientation · `A=012021021 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:2 d6:1 d7:0 d8:2]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] 1 colouring(s) with down-apex sequence `012012`: -- face apexes (raw labels) `102102` → canonical `012012` · `A=012021021 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102102`, realises `012012` in some orientation · `A=012021021 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:2 d6:1 d7:0 d8:2]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] 1 colouring(s) with down-apex sequence `012012`: -- face apexes (raw labels) `201201` → canonical `012012` · `A=010212012 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201201`, realises `012012` in some orientation · `A=010212012 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:1 d6:2 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.pdf index cc7cbef..477391f 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012012.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.md index 90fbf18..608b9f4 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.md @@ -1,17 +1,38 @@ # Inner-face down-apex sequence `012021` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **1** of 7 configs (M(T), inner face). -- **1** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **4** of 7 configs (M(T), inner face). +- **7** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_012021.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. -## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] 1 colouring(s) with down-apex sequence `012021`: -- face apexes (raw labels) `120102` → canonical `012021` · `A=012012021 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `021201`, realises `012021` in some orientation · `A=012102012 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:2 d7:0 d8:1]` + +## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] + +2 colouring(s) with down-apex sequence `012021`: + +- face apex colours (canonical-rep edge order) `120102`, realises `012021` in some orientation · `A=012012021 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `021201`, realises `012021` in some orientation · `A=012102012 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:2 d7:0 d8:1]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +3 colouring(s) with down-apex sequence `012021`: + +- face apex colours (canonical-rep edge order) `021201`, realises `012021` in some orientation · `A=012012012 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `012102`, realises `012021` in some orientation · `A=012021021 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `021201`, realises `012021` in some orientation · `A=012102012 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +1 colouring(s) with down-apex sequence `012021`: + +- face apex colours (canonical-rep edge order) `021201`, realises `012021` in some orientation · `A=012102012 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:2 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.pdf index fac8ef0..93be6fa 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.png index 242eb30..0731dc4 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012021.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.md index e8d53df..a1b367b 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.md @@ -1,10 +1,10 @@ # Inner-face down-apex sequence `012102` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. - Realised by **4** of 7 configs (M(T), inner face). -- **6** Kempe-balanced colourings (mod colour permutation) produce it. +- **7** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_012102.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. @@ -13,25 +13,26 @@ Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D 1 colouring(s) with down-apex sequence `012102`: -- face apexes (raw labels) `021201` → canonical `012102` · `A=012102012 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `021201`, realises `012102` in some orientation · `A=012102012 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:2 d7:0 d8:1]` ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `012102`: +2 colouring(s) with down-apex sequence `012102`: -- face apexes (raw labels) `021201` → canonical `012102` · `A=012102012 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `120102`, realises `012102` in some orientation · `A=012012021 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `021201`, realises `012102` in some orientation · `A=012102012 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:2 d7:0 d8:1]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] 3 colouring(s) with down-apex sequence `012102`: -- face apexes (raw labels) `021201` → canonical `012102` · `A=012012012 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `012102` → canonical `012102` · `A=012021021 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `021201` → canonical `012102` · `A=012102012 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `021201`, realises `012102` in some orientation · `A=012012012 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `012102`, realises `012102` in some orientation · `A=012021021 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `021201`, realises `012102` in some orientation · `A=012102012 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:2 d7:0 d8:1]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] 1 colouring(s) with down-apex sequence `012102`: -- face apexes (raw labels) `021201` → canonical `012102` · `A=012102012 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `021201`, realises `012102` in some orientation · `A=012102012 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:2 d7:0 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.pdf index c0a32dc..30a5da6 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.png index c3ee737..e31d205 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012102.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.md index 4a6e26c..3e52278 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.md @@ -1,23 +1,53 @@ # Inner-face down-apex sequence `012120` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **2** of 7 configs (M(T), inner face). -- **2** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **5** of 7 configs (M(T), inner face). +- **17** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_012120.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + +2 colouring(s) with down-apex sequence `012120`: + +- face apex colours (canonical-rep edge order) `021102`, realises `012120` in some orientation · `A=012102021 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `012120` in some orientation · `A=012102102 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]` + ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `012120`: +2 colouring(s) with down-apex sequence `012120`: -- face apexes (raw labels) `120201` → canonical `012120` · `A=012012012 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `120201`, realises `012120` in some orientation · `A=012012012 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `021102`, realises `012120` in some orientation · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:1 d7:0 d8:2]` + +## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] + +6 colouring(s) with down-apex sequence `012120`: + +- face apex colours (canonical-rep edge order) `021102`, realises `012120` in some orientation · `A=012012021 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `012120` in some orientation · `A=012012102 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `012201`, realises `012120` in some orientation · `A=012021012 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `012120` in some orientation · `A=012021201 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `012120` in some orientation · `A=012102021 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `012120` in some orientation · `A=012102102 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + +4 colouring(s) with down-apex sequence `012120`: + +- face apex colours (canonical-rep edge order) `212001`, realises `012120` in some orientation · `A=010201212 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020211`, realises `012120` in some orientation · `A=012012102 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `012120` in some orientation · `A=012021201 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `012120` in some orientation · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:1 d7:0 d8:2]` ## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] -1 colouring(s) with down-apex sequence `012120`: +3 colouring(s) with down-apex sequence `012120`: -- face apexes (raw labels) `210102` → canonical `012120` · `A=010212021 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:0 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `212001`, realises `012120` in some orientation · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `210102`, realises `012120` in some orientation · `A=010212021 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:0 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `011202`, realises `012120` in some orientation · `A=012020121 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.pdf index daf30fd..9685c7e 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.png index 3460648..cc2a3cb 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012120.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.md index 9791341..447d7e3 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.md @@ -1,37 +1,53 @@ # Inner-face down-apex sequence `012201` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **4** of 7 configs (M(T), inner face). -- **6** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **5** of 7 configs (M(T), inner face). +- **17** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_012201.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] -1 colouring(s) with down-apex sequence `012201`: +2 colouring(s) with down-apex sequence `012201`: -- face apexes (raw labels) `021102` → canonical `012201` · `A=012102021 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `012201` in some orientation · `A=012102021 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `012201` in some orientation · `A=012102102 U[u0:2 u1:0 u4:1] D[d2:0 d3:2 d5:0 d6:2 d7:1 d8:1]` ## C03 — word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `012201`: +2 colouring(s) with down-apex sequence `012201`: -- face apexes (raw labels) `021102` → canonical `012201` · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `120201`, realises `012201` in some orientation · `A=012012012 U[u0:2 u1:0 u5:1] D[d2:1 d3:2 d4:0 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `021102`, realises `012201` in some orientation · `A=012102021 U[u0:2 u1:0 u5:1] D[d2:0 d3:2 d4:1 d6:1 d7:0 d8:2]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] -3 colouring(s) with down-apex sequence `012201`: +6 colouring(s) with down-apex sequence `012201`: -- face apexes (raw labels) `021102` → canonical `012201` · `A=012012021 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `012201` → canonical `012201` · `A=012021012 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:2 d7:0 d8:1]` -- face apexes (raw labels) `021102` → canonical `012201` · `A=012102021 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `012201` in some orientation · `A=012012021 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `012201` in some orientation · `A=012012102 U[u0:2 u2:1 u4:0] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `012201`, realises `012201` in some orientation · `A=012021012 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `012201` in some orientation · `A=012021201 U[u0:2 u2:1 u4:0] D[d1:0 d3:1 d5:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `012201` in some orientation · `A=012102021 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `020211`, realises `012201` in some orientation · `A=012102102 U[u0:2 u2:0 u4:1] D[d1:0 d3:2 d5:0 d6:2 d7:1 d8:1]` ## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] -1 colouring(s) with down-apex sequence `012201`: +4 colouring(s) with down-apex sequence `012201`: -- face apexes (raw labels) `021102` → canonical `012201` · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `212001`, realises `012201` in some orientation · `A=010201212 U[u0:2 u2:1 u5:0] D[d1:2 d3:1 d4:2 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `020211`, realises `012201` in some orientation · `A=012012102 U[u0:2 u2:1 u5:0] D[d1:0 d3:2 d4:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `010122`, realises `012201` in some orientation · `A=012021201 U[u0:2 u2:1 u5:0] D[d1:0 d3:1 d4:0 d6:1 d7:2 d8:2]` +- face apex colours (canonical-rep edge order) `021102`, realises `012201` in some orientation · `A=012102021 U[u0:2 u2:0 u5:1] D[d1:0 d3:2 d4:1 d6:1 d7:0 d8:2]` + +## C06 — word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] + +3 colouring(s) with down-apex sequence `012201`: + +- face apex colours (canonical-rep edge order) `212001`, realises `012201` in some orientation · `A=010201212 U[u0:2 u3:1 u6:0] D[d1:2 d2:1 d4:2 d5:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `210102`, realises `012201` in some orientation · `A=010212021 U[u0:2 u3:0 u6:1] D[d1:2 d2:1 d4:0 d5:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `011202`, realises `012201` in some orientation · `A=012020121 U[u0:2 u3:1 u6:0] D[d1:0 d2:1 d4:1 d5:2 d7:0 d8:2]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.pdf index 74b1642..0d87ab8 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.png index ac8a5b2..f02bc49 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012201.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.md index 87efeb3..8c31e80 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.md @@ -1,30 +1,49 @@ # Inner-face down-apex sequence `012210` -Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in increasing annular-edge order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. +Canonical colour sequence of the singleton down-tooth apexes on a single inner non-tooth face (read in cyclic order, reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 singleton down apexes on the face**. Sequences are read off the un-deduped census (every cyclic orientation of each colouring), so one colouring may realise several sequences -- see `../kempe_sequence_orientation_note.md`. - Colour multiset: 2×colour0, 2×colour1, 2×colour2. -- Realised by **3** of 7 configs (M(T), inner face). -- **4** Kempe-balanced colourings (mod colour permutation) produce it. +- Realised by **5** of 7 configs (M(T), inner face). +- **13** Kempe-balanced colourings (mod colour permutation) realise it in some orientation. - Figure: `seq_012210.png` (black rings mark the face's down apexes). Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours 0/1/2 = 0:orange, 1:blue, 2:green. ## C00 — word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] -2 colouring(s) with down-apex sequence `012210`: +6 colouring(s) with down-apex sequence `012210`: -- face apexes (raw labels) `201102` → canonical `012210` · `A=012012021 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:1 d6:1 d7:0 d8:2]` -- face apexes (raw labels) `102201` → canonical `012210` · `A=012021012 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `221001`, realises `012210` in some orientation · `A=012010212 U[u0:2 u1:0 u2:1] D[d3:2 d4:2 d5:1 d6:0 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `201102`, realises `012210` in some orientation · `A=012012021 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `200211`, realises `012210` in some orientation · `A=012012102 U[u0:2 u1:0 u2:1] D[d3:2 d4:0 d5:0 d6:2 d7:1 d8:1]` +- face apex colours (canonical-rep edge order) `112002`, realises `012210` in some orientation · `A=012020121 U[u0:2 u1:0 u2:1] D[d3:1 d4:1 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102201`, realises `012210` in some orientation · `A=012021012 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100122`, realises `012210` in some orientation · `A=012021201 U[u0:2 u1:0 u2:1] D[d3:1 d4:0 d5:0 d6:1 d7:2 d8:2]` ## C01 — word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] +3 colouring(s) with down-apex sequence `012210`: + +- face apex colours (canonical-rep edge order) `112002`, realises `012210` in some orientation · `A=012020121 U[u0:2 u1:0 u3:1] D[d2:1 d4:1 d5:2 d6:0 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `102201`, realises `012210` in some orientation · `A=012021012 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `100122`, realises `012210` in some orientation · `A=012021201 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:0 d6:1 d7:2 d8:2]` + +## C02 — word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] + 1 colouring(s) with down-apex sequence `012210`: -- face apexes (raw labels) `102201` → canonical `012210` · `A=012021012 U[u0:2 u1:0 u3:1] D[d2:1 d4:0 d5:2 d6:2 d7:0 d8:1]` +- face apex colours (canonical-rep edge order) `112002`, realises `012210` in some orientation · `A=012020121 U[u0:2 u1:0 u4:1] D[d2:1 d3:1 d5:2 d6:0 d7:0 d8:2]` ## C04 — word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] +2 colouring(s) with down-apex sequence `012210`: + +- face apex colours (canonical-rep edge order) `201102`, realises `012210` in some orientation · `A=010212021 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `200211`, realises `012210` in some orientation · `A=010212102 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:0 d6:2 d7:1 d8:1]` + +## C05 — word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] + 1 colouring(s) with down-apex sequence `012210`: -- face apexes (raw labels) `201102` → canonical `012210` · `A=010212021 U[u0:2 u2:1 u4:0] D[d1:2 d3:0 d5:1 d6:1 d7:0 d8:2]` +- face apex colours (canonical-rep edge order) `200211`, realises `012210` in some orientation · `A=010212102 U[u0:2 u2:1 u5:0] D[d1:2 d3:0 d4:0 d6:2 d7:1 d8:1]` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.pdf index 1f7451d..fa2ed9a 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.pdf and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.pdf differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.png b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.png index a8a3e96..5299185 100644 Binary files a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.png and b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/seq_012210.png differ diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/summary.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/summary.md index d7b0789..28128c4 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/summary.md +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/summary.md @@ -1,42 +1,45 @@ # Inner-face singleton-down-apex sequences of Kempe-balanced colourings (n=9, m=6) -Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 6 singleton down-tooth apexes: **7 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in increasing annular-edge order, and reduce it modulo colour permutation (NOT dihedral symmetry). +Every full medial tire graph M(T) with |A(T)| = 9 (one representative per dihedral class) that has an inner non-tooth face holding exactly 6 singleton down-tooth apexes: **7 configs (M(T), inner face)**. For each we enumerate the Kempe-balanced (valid) proper 3-colourings (modulo colour permutation), read the down-apex colour sequence in cyclic order off the un-deduped census (every cyclic orientation of each colouring), and reduce it modulo colour permutation (NOT dihedral symmetry). Reading off the census makes the recorded vocabulary orientation-honest; see `../kempe_sequence_orientation_note.md`. - Total Kempe-balanced colourings (mod colour permutation): **127**. -- Distinct canonical down-apex sequences overall: **28**. +- Distinct canonical down-apex sequences overall: **31**. ## Distinct canonical down-apex sequences | sequence | colour multiset | #configs realising | #colourings | |---|---|---|---| | `000000` | 6 | 4 | 5 | -| `000011` | 4+2 | 5 | 7 | -| `000101` | 4+2 | 1 | 1 | -| `000110` | 4+2 | 4 | 5 | -| `001001` | 4+2 | 4 | 6 | -| `001010` | 4+2 | 1 | 1 | -| `001100` | 4+2 | 4 | 6 | -| `001111` | 4+2 | 7 | 12 | -| `001122` | 2+2+2 | 5 | 8 | -| `001221` | 2+2+2 | 3 | 4 | -| `010001` | 4+2 | 4 | 7 | -| `010100` | 4+2 | 2 | 2 | -| `010111` | 4+2 | 4 | 8 | -| `010122` | 2+2+2 | 3 | 6 | -| `010221` | 2+2+2 | 2 | 2 | -| `011000` | 4+2 | 5 | 7 | -| `011011` | 4+2 | 2 | 2 | -| `011022` | 2+2+2 | 4 | 5 | -| `011101` | 4+2 | 1 | 1 | -| `011110` | 4+2 | 4 | 5 | -| `011202` | 2+2+2 | 1 | 1 | -| `011220` | 2+2+2 | 3 | 3 | +| `000011` | 4+2 | 7 | 42 | +| `000101` | 4+2 | 4 | 20 | +| `000110` | 4+2 | 7 | 42 | +| `001001` | 4+2 | 4 | 8 | +| `001010` | 4+2 | 4 | 20 | +| `001100` | 4+2 | 7 | 42 | +| `001111` | 4+2 | 7 | 42 | +| `001122` | 2+2+2 | 5 | 11 | +| `001212` | 2+2+2 | 5 | 17 | +| `001221` | 2+2+2 | 5 | 13 | +| `010001` | 4+2 | 4 | 20 | +| `010010` | 4+2 | 4 | 8 | +| `010100` | 4+2 | 4 | 20 | +| `010111` | 4+2 | 4 | 20 | +| `010122` | 2+2+2 | 5 | 17 | +| `010212` | 2+2+2 | 4 | 7 | +| `010221` | 2+2+2 | 5 | 17 | +| `011000` | 4+2 | 7 | 42 | +| `011011` | 4+2 | 4 | 8 | +| `011022` | 2+2+2 | 5 | 13 | +| `011101` | 4+2 | 4 | 20 | +| `011110` | 4+2 | 7 | 42 | +| `011202` | 2+2+2 | 5 | 17 | +| `011220` | 2+2+2 | 5 | 11 | | `012012` | 2+2+2 | 3 | 4 | -| `012021` | 2+2+2 | 1 | 1 | -| `012102` | 2+2+2 | 4 | 6 | -| `012120` | 2+2+2 | 2 | 2 | -| `012201` | 2+2+2 | 4 | 6 | -| `012210` | 2+2+2 | 3 | 4 | +| `012021` | 2+2+2 | 4 | 7 | +| `012102` | 2+2+2 | 4 | 7 | +| `012120` | 2+2+2 | 5 | 17 | +| `012201` | 2+2+2 | 5 | 17 | +| `012210` | 2+2+2 | 5 | 13 | Note: every realised sequence has its three colour-counts of **equal parity** — exactly the Kempe-parity constraint on the inner face (each colour pair meets its singleton down apexes an even number of times). With m = 6 apexes (m is even) every count must be **even**, so the only admissible colour multisets are 2+2+2, 4+2, 6. @@ -47,11 +50,11 @@ The 7 configs fall into **6** groups by the set of canonical down-apex sequences | #configs | set of down-apex sequences | config ids | |---|---|---| | 2 | { `000000`, `000011`, `000110`, `001100`, `001111`, `001221`, `011000`, `011022`, `011110`, `012012`, `012210` } | C00, C01 | -| 1 | { `000000`, `000011`, `000110`, `001001`, `001100`, `001111`, `001122`, `001221`, `010001`, `010111`, `010122`, `012102`, `012201` } | C02 | -| 1 | { `000000`, `000011`, `000110`, `001001`, `001111`, `001122`, `011000`, `011011`, `011220`, `012021`, `012102`, `012120`, `012201` } | C03 | -| 1 | { `001001`, `001111`, `001122`, `010001`, `010111`, `010122`, `011000`, `011022`, `011110`, `012012`, `012102`, `012201`, `012210` } | C04 | -| 1 | { `000011`, `000101`, `001010`, `001100`, `001111`, `001122`, `010001`, `010100`, `010111`, `010221`, `011101`, `011202`, `011220`, `012120` } | C06 | -| 1 | { `001001`, `001111`, `001122`, `010001`, `010100`, `010111`, `010122`, `010221`, `011000`, `011011`, `011022`, `011110`, `011220`, `012102`, `012201` } | C05 | +| 1 | { `000011`, `000101`, `000110`, `001010`, `001100`, `001111`, `001122`, `001212`, `010001`, `010100`, `010111`, `010122`, `010221`, `011000`, `011101`, `011110`, `011202`, `011220`, `012120`, `012201` } | C06 | +| 1 | { `000000`, `000011`, `000110`, `001001`, `001100`, `001111`, `001122`, `001212`, `010010`, `010122`, `010212`, `010221`, `011000`, `011011`, `011110`, `011202`, `011220`, `012021`, `012102`, `012120`, `012201` } | C03 | +| 1 | { `000011`, `000101`, `000110`, `001001`, `001010`, `001100`, `001111`, `001122`, `001212`, `001221`, `010001`, `010010`, `010100`, `010111`, `010122`, `010212`, `010221`, `011000`, `011011`, `011022`, `011101`, `011110`, `011202`, `011220`, `012021`, `012102`, `012120`, `012201`, `012210` } | C05 | +| 1 | { `000000`, `000011`, `000101`, `000110`, `001001`, `001010`, `001100`, `001111`, `001122`, `001212`, `001221`, `010001`, `010010`, `010100`, `010111`, `010122`, `010212`, `010221`, `011000`, `011011`, `011022`, `011101`, `011110`, `011202`, `011220`, `012021`, `012102`, `012120`, `012201`, `012210` } | C02 | +| 1 | { `000011`, `000101`, `000110`, `001001`, `001010`, `001100`, `001111`, `001122`, `001212`, `001221`, `010001`, `010010`, `010100`, `010111`, `010122`, `010212`, `010221`, `011000`, `011011`, `011022`, `011101`, `011110`, `011202`, `011220`, `012012`, `012021`, `012102`, `012120`, `012201`, `012210` } | C04 | ## Config atlas (ids) @@ -59,11 +62,11 @@ The 7 configs fall into **6** groups by the set of canonical down-apex sequences |---|---|---|---| | C00 | word=UUUDDDDDD bites=- face=root apexes=[d3,d4,d5,d6,d7,d8] | 22 | { `000000`, `000011`, `000110`, `001100`, `001111`, `001221`, `011000`, `011022`, `011110`, `012012`, `012210` } | | C01 | word=UUDUDDDDD bites=- face=root apexes=[d2,d4,d5,d6,d7,d8] | 11 | { `000000`, `000011`, `000110`, `001100`, `001111`, `001221`, `011000`, `011022`, `011110`, `012012`, `012210` } | -| C02 | word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] | 17 | { `000000`, `000011`, `000110`, `001001`, `001100`, `001111`, `001122`, `001221`, `010001`, `010111`, `010122`, `012102`, `012201` } | -| C03 | word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] | 13 | { `000000`, `000011`, `000110`, `001001`, `001111`, `001122`, `011000`, `011011`, `011220`, `012021`, `012102`, `012120`, `012201` } | -| C04 | word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] | 26 | { `001001`, `001111`, `001122`, `010001`, `010111`, `010122`, `011000`, `011022`, `011110`, `012012`, `012102`, `012201`, `012210` } | -| C05 | word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] | 20 | { `001001`, `001111`, `001122`, `010001`, `010100`, `010111`, `010122`, `010221`, `011000`, `011011`, `011022`, `011110`, `011220`, `012102`, `012201` } | -| C06 | word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] | 18 | { `000011`, `000101`, `001010`, `001100`, `001111`, `001122`, `010001`, `010100`, `010111`, `010221`, `011101`, `011202`, `011220`, `012120` } | +| C02 | word=UUDDUDDDD bites=- face=root apexes=[d2,d3,d5,d6,d7,d8] | 17 | { `000000`, `000011`, `000101`, `000110`, `001001`, `001010`, `001100`, `001111`, `001122`, `001212`, `001221`, `010001`, `010010`, `010100`, `010111`, `010122`, `010212`, `010221`, `011000`, `011011`, `011022`, `011101`, `011110`, `011202`, `011220`, `012021`, `012102`, `012120`, `012201`, `012210` } | +| C03 | word=UUDDDUDDD bites=- face=root apexes=[d2,d3,d4,d6,d7,d8] | 13 | { `000000`, `000011`, `000110`, `001001`, `001100`, `001111`, `001122`, `001212`, `010010`, `010122`, `010212`, `010221`, `011000`, `011011`, `011110`, `011202`, `011220`, `012021`, `012102`, `012120`, `012201` } | +| C04 | word=UDUDUDDDD bites=- face=root apexes=[d1,d3,d5,d6,d7,d8] | 26 | { `000011`, `000101`, `000110`, `001001`, `001010`, `001100`, `001111`, `001122`, `001212`, `001221`, `010001`, `010010`, `010100`, `010111`, `010122`, `010212`, `010221`, `011000`, `011011`, `011022`, `011101`, `011110`, `011202`, `011220`, `012012`, `012021`, `012102`, `012120`, `012201`, `012210` } | +| C05 | word=UDUDDUDDD bites=- face=root apexes=[d1,d3,d4,d6,d7,d8] | 20 | { `000011`, `000101`, `000110`, `001001`, `001010`, `001100`, `001111`, `001122`, `001212`, `001221`, `010001`, `010010`, `010100`, `010111`, `010122`, `010212`, `010221`, `011000`, `011011`, `011022`, `011101`, `011110`, `011202`, `011220`, `012021`, `012102`, `012120`, `012201`, `012210` } | +| C06 | word=UDDUDDUDD bites=- face=root apexes=[d1,d2,d4,d5,d7,d8] | 18 | { `000011`, `000101`, `000110`, `001010`, `001100`, `001111`, `001122`, `001212`, `010001`, `010100`, `010111`, `010122`, `010221`, `011000`, `011101`, `011110`, `011202`, `011220`, `012120`, `012201` } | ## Per-sequence notes @@ -76,11 +79,14 @@ The 7 configs fall into **6** groups by the set of canonical down-apex sequences - [`001100`](seq_001100.md) — figure `seq_001100.png` - [`001111`](seq_001111.md) — figure `seq_001111.png` - [`001122`](seq_001122.md) — figure `seq_001122.png` +- [`001212`](seq_001212.md) — figure `seq_001212.png` - [`001221`](seq_001221.md) — figure `seq_001221.png` - [`010001`](seq_010001.md) — figure `seq_010001.png` +- [`010010`](seq_010010.md) — figure `seq_010010.png` - [`010100`](seq_010100.md) — figure `seq_010100.png` - [`010111`](seq_010111.md) — figure `seq_010111.png` - [`010122`](seq_010122.md) — figure `seq_010122.png` +- [`010212`](seq_010212.md) — figure `seq_010212.png` - [`010221`](seq_010221.md) — figure `seq_010221.png` - [`011000`](seq_011000.md) — figure `seq_011000.png` - [`011011`](seq_011011.md) — figure `seq_011011.png` diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_sequence_orientation_note.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_sequence_orientation_note.md new file mode 100644 index 0000000..3fe72e9 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_sequence_orientation_note.md @@ -0,0 +1,73 @@ +# Reading apex sequences off the un-deduped census + +This note documents a convention used by `kempe_up_tooth_sequences.py` and +`kempe_down_face_sequences.py`, and the artifact it fixes. + +## The artifact + +Both experiments record, for each Kempe-balanced colouring, the colour sequence +of a distinguished apex set (up-tooth apexes on the outer face; singleton +down-tooth apexes on one inner face), read **in cyclic order around the annular +cycle** and reduced **modulo colour permutation only — NOT dihedral symmetry** +(per the experiment's step 3). + +The graphs, however, *are* deduplicated by the full dihedral group of the cycle +(rotations and reflections), keeping one canonical representative per class. +These two conventions disagree about the cycle's symmetry: + +- graphs: quotiented by rotation **and** reflection; +- sequences: read at a fixed anchor (edge 0), quotiented by neither. + +The apex reading is not invariant under the rotations/reflections we quotient +the graphs by. So reading off a single canonical representative records only the +one orientation of each necklace that happens to align with that representative +— and which orientation that is depends on the lexicographic tie-break used to +pick the representative, which has nothing to do with the colouring. + +### Symptom (n = 9, m = 6, outer vs inner) + +The up-tooth (outer-face) and down-face (inner-face) experiments at m = 6 run on +graph populations that are exact U↔D embedding mirrors of each other, so every +honest aggregate should match. Under the anchored reading they did *not*: each +realised 28 sequences, but they differed by one element — up uniquely realised +`001212`, down uniquely realised `010122`. Those two strings are the same +necklace (one is a rotation+recolour of the other); the split was purely an +artifact of the two experiments anchoring their canonical representatives +differently. + +Reading off the **full un-deduped census** instead, each side realises 31 +sequences and the two vocabularies are **identical**. + +## The fix + +`dihedral_reading_sequences(n, coloring, edges, prefix)` reads the apex sequence +from *every* anchor in the cycle's dihedral group — applying each symmetry +`i -> (a*i + b) mod n` (`a in {1,-1}`, `b in 0..n-1`, the same group +`canonical_key()` quotients graphs by) to the edge positions and collecting the +canonical (colour-permutation) reading from each. The union over all `2n` +anchors equals exactly the set of sequences the brute un-deduped census would +record for that colouring's orbit (verified by direct comparison). + +This is the **default** reading for both scripts. Consequences: + +- The recorded vocabulary is orientation-honest: it is closed under the same + symmetry the graph census is, so outer/inner (and any rotation/reflection of a + configuration) agree. +- A single colouring may now realise several sequences (its whole orbit of + anchored readings), so a colouring can appear under more than one + per-sequence note, and the per-sequence colouring counts sum to more than the + number of distinct colourings. +- The colour-count **parity law is unaffected** — rotating/reflecting a reading + permutes positions but not the colour multiset. + +The old single-anchor behaviour is still available via `--anchored` on either +script, for reproducing the earlier (artifact-bearing) runs. + +## Note on necklaces vs. orientations + +This is *not* the same as quotienting sequences by rotation/reflection +(necklaces). The apexes occupy a subset of the `n` edge positions, so the +dihedral anchors generate only those cyclic cuts the actual cycle symmetry +permits — at n = 9, m = 6 that is 31 distinct strings, not the 9 full necklaces. +The census keeps genuinely distinct linear sequences; it just stops privileging +one arbitrary anchor. diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences.py b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences.py index a0bb3cc..c61bc20 100644 --- a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences.py +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences.py @@ -72,6 +72,26 @@ def _parity_partitions(m: int) -> set[tuple[int, ...]]: return out +def dihedral_reading_sequences( + n: int, coloring: Coloring, edges, prefix: str +) -> set[tuple[int, ...]]: + """Every canonical apex reading over the n-cycle's dihedral symmetries. + + The apexes on ``edges`` are read in cyclic order; reading off a single fixed + representative anchors that order arbitrarily at edge 0. Applying each + dihedral symmetry i -> (a*i + b) mod n (a in {1,-1}) to the edge positions -- + the same group canonical_key() quotients graphs by -- and collecting the + canonical reading from each anchor reproduces exactly the orientation-honest + *un-deduped census* vocabulary (verified: it equals the brute census). + """ + out: set[tuple[int, ...]] = set() + for a in (1, -1): + for b in range(n): + order = sorted(edges, key=lambda e: (a * e + b) % n) + out.add(canonical_sequence(tuple(coloring[f"{prefix}{e}"] for e in order))) + return out + + def up_tooth_sequence(graph: FullMedialTireGraph, coloring: Coloring) -> tuple[int, ...]: """Colours of the up-tooth apexes in increasing annular-edge order.""" return tuple(coloring[f"u{i}"] for i in graph.up_edges) @@ -91,29 +111,34 @@ def describe_graph(graph: FullMedialTireGraph) -> str: # --------------------------------------------------------------------------- class Experiment: - def __init__(self, n: int, m: int): + def __init__(self, n: int, m: int, dihedral: bool = True): self.n = n self.m = m + self.dihedral = dihedral # read sequences off the un-deduped census self.graphs: list[FullMedialTireGraph] = [ g for g in generate(n, min_up_teeth=3, dedup=True) if len(g.up_edges) == m ] - # per graph: list of (coloring, canonical up-tooth sequence) - self.colorings: list[list[tuple[Coloring, tuple[int, ...]]]] = [] + # per graph: list of (coloring, set of canonical up-tooth sequences) + self.colorings: list[list[tuple[Coloring, frozenset]]] = [] # per graph: set of canonical up-tooth sequences it realises self.graph_seq_sets: list[frozenset[tuple[int, ...]]] = [] # canonical sequence -> list of (graph_idx, coloring) self.by_sequence: dict[tuple[int, ...], list[tuple[int, Coloring]]] = defaultdict(list) for gidx, g in enumerate(self.graphs): - entries: list[tuple[Coloring, tuple[int, ...]]] = [] + entries: list[tuple[Coloring, frozenset]] = [] seqs: set[tuple[int, ...]] = set() for coloring, verdict in classify_colorings(g, dedup_colors=True): if not verdict.valid: continue - cseq = canonical_sequence(up_tooth_sequence(g, coloring)) - entries.append((coloring, cseq)) - seqs.add(cseq) - self.by_sequence[cseq].append((gidx, coloring)) + if self.dihedral: + cseqs = dihedral_reading_sequences(g.n, coloring, g.up_edges, "u") + else: + cseqs = {canonical_sequence(up_tooth_sequence(g, coloring))} + entries.append((coloring, frozenset(cseqs))) + for cseq in cseqs: + seqs.add(cseq) + self.by_sequence[cseq].append((gidx, coloring)) self.colorings.append(entries) self.graph_seq_sets.append(frozenset(seqs)) @@ -251,17 +276,19 @@ def write_sequence_note(exp: Experiment, seq, path, fig_name): lines.append(f"# Up-tooth apex sequence `{s}`") lines.append("") lines.append( - f"Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic " - f"order around A(T), reduced modulo the six colour permutations) for " - f"Kempe-balanced 3-colourings of M(T) with **n = {exp.n}**, " - f"**m = {exp.m} up teeth**." + f"Canonical up-tooth apex colour sequence (the apexes u0, u1, ... read in " + f"cyclic order around A(T), reduced modulo the six colour permutations) " + f"for Kempe-balanced 3-colourings of M(T) with **n = {exp.n}**, " + f"**m = {exp.m} up teeth**. Sequences are read off the un-deduped census " + f"(every cyclic orientation of each colouring), so one colouring may " + f"realise several sequences -- see `../kempe_sequence_orientation_note.md`." ) lines.append("") lines.append(f"- Colour multiset: {counts}.") lines.append(f"- Realised by **{len(by_graph)}** of {len(exp.graphs)} M(T) " f"(dihedral classes).") lines.append(f"- **{len(exp.by_sequence[seq])}** Kempe-balanced colourings " - f"(mod colour permutation) produce it.") + f"(mod colour permutation) realise it in some orientation.") lines.append(f"- Figure: `{fig_name}` (black rings mark the up-tooth apexes).") lines.append("") lines.append("Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` " @@ -278,7 +305,8 @@ def write_sequence_note(exp: Experiment, seq, path, fig_name): lines.append("") for coloring in cols: raw = seq_str(up_tooth_sequence(g, coloring)) - lines.append(f"- up apexes (raw labels) `{raw}` → canonical `{s}` · " + lines.append(f"- apex colours (canonical-rep edge order) `{raw}`, " + f"realises `{s}` in some orientation · " f"`{compact_coloring(g, coloring)}`") lines.append("") with open(path, "w") as fh: @@ -291,13 +319,18 @@ def write_summary(exp: Experiment, path, notes_dir_name): lines.append(f"# Up-tooth apex sequences of Kempe-balanced colourings " f"(n={exp.n}, m={exp.m})") lines.append("") + reading = ("the un-deduped census (every cyclic orientation of each " + "colouring)" if exp.dihedral + else "a single anchored representative per dihedral class") lines.append( f"Every full medial tire graph M(T) with |A(T)| = {exp.n} and exactly " f"{exp.m} up teeth, one representative per dihedral class: " f"**{len(exp.graphs)} M(T)**. For each we enumerate the Kempe-balanced " f"(valid) proper 3-colourings (modulo colour permutation), read the " - f"up-tooth apex colour sequence u0