From d8b5975f817032b0db8b723a935de4b32067143a Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 11 Jun 2026 21:19:35 -0400 Subject: [PATCH] Add inner-face down-apex colour-sequence experiment (n=9 sweep) Mirror of the up-tooth experiment with the distinguished valid face moved from the outer face to an inner non-tooth face (root or bite inner-gap). For each (M(T), inner face) config holding m singleton down-tooth apexes, record the apex colour sequence (cyclic order, mod colour permutation) over Kempe-balanced colourings and group configs by their sequence-set. Runs for m=3,4,5,6 with per-sequence notes, figures, and a config atlas. Finding: inner faces realise the same parity-admissible sequence vocabulary and the same distinct-sequence counts (1/4/10/28) as the outer face, i.e. the Kempe-parity law acts uniformly on every valid face. At m=6 the configs are the U<->D embedding mirror of the up-m=6 graphs (matching 7 configs, 28 sequences, 127 colourings). 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papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m6/summary.md 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 new file mode 100644 index 0000000..81abca2 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences.py @@ -0,0 +1,384 @@ +"""Singleton-down-apex colour sequences of Kempe-balanced 3-colourings. + +Inner-face counterpart of ``kempe_up_tooth_sequences.py``. There the role of +the distinguished valid face was the unique *outer* face, which carries every +up-tooth apex; here we play the same game on an *inner* non-tooth face of B(T) +(the root face, or a bite's inner-gap face), which carries the singleton +down-tooth apexes assigned to it. + +For a fixed annular size ``n`` and a fixed count ``m`` we: + + 1. take every full medial tire graph M(T) with |A(T)| = n (one representative + per dihedral symmetry class) that has an inner non-tooth face F holding + exactly ``m`` singleton down-tooth apexes -- by Remark 3.8 every such face + holds 0 or >= 3, so m >= 3. (At n = 9 each M(T) has at most one inner face + bearing singletons, so a graph and its face coincide one-to-one.) + 2. enumerate the Kempe-balanced (``valid``) proper 3-colourings of M(T) and + read off the colour sequence of F's singleton down-tooth apexes d_i in + increasing annular-edge order (their cyclic order along F's arc); + 3. reduce each sequence modulo the six colour permutations -- NOT modulo the + dihedral symmetry of the cycle -- to a canonical sequence; + 4. group the configurations by the *set* of canonical down-apex sequences + they realise, and report how many share each set. + +The Kempe-balanced rule is global, so the same valid colourings are used as in +the up-tooth experiment; only the apex set we read changes. Note that the rule +forces, on every valid face, each colour pair to meet the counted apexes an even +number of times -- so the down-apex sequences obey the same equal-parity law as +the up-tooth sequences. + +Run: python3 kempe_down_face_sequences.py --n 9 --m 3 +""" + +from __future__ import annotations + +import argparse +import math +import os +from collections import defaultdict + +from full_medial_tire_generator import ( + FullMedialTireGraph, + generate, + innermost_bite, +) +from kempe_valid_colorings import classify_colorings +from kempe_up_tooth_sequences import ( + PALETTE, + PALETTE_NAME, + _parity_partitions, + _positions, + canonical_sequence, + compact_coloring, + seq_str, +) + +HERE = os.path.dirname(os.path.abspath(__file__)) +Coloring = dict[str, int] + +Bite = tuple[int, int] +FaceKey = Bite | None # None = root face + + +def face_name(face: FaceKey) -> str: + return "root" if face is None else f"bite({face[0]},{face[1]})" + + +def inner_face_singletons(graph: FullMedialTireGraph) -> dict[FaceKey, list[int]]: + """Map each inner non-tooth face to its singleton down-tooth edges (sorted).""" + perface: dict[FaceKey, list[int]] = defaultdict(list) + for e in graph.singleton_down_edges: + perface[innermost_bite(e, graph.bites)].append(e) + return {face: sorted(edges) for face, edges in perface.items()} + + +def config_id(idx: int) -> str: + return f"C{idx:02d}" + + +def describe_config(graph: FullMedialTireGraph, face: FaceKey, edges) -> str: + bites = ",".join(f"({i},{j})" for i, j in sorted(graph.bites)) or "-" + apexes = ",".join(f"d{e}" for e in edges) + return (f"word={graph.tooth_word} bites={bites} " + f"face={face_name(face)} apexes=[{apexes}]") + + +# --------------------------------------------------------------------------- +# Data collection. +# --------------------------------------------------------------------------- + +class Config: + """A single (M(T), inner face) specimen with m singleton down apexes.""" + + def __init__(self, graph: FullMedialTireGraph, face: FaceKey, edges: list[int]): + self.graph = graph + self.face = face + self.edges = edges # singleton down edges on the face + self.apexes = [f"d{e}" for e in edges] # their apex vertex names + + def sequence(self, coloring: Coloring) -> tuple[int, ...]: + return tuple(coloring[v] for v in self.apexes) + + +class Experiment: + def __init__(self, n: int, m: int): + self.n = n + self.m = m + self.configs: list[Config] = [] + for g in generate(n, min_up_teeth=3, dedup=True): + for face, edges in inner_face_singletons(g).items(): + if len(edges) == m: + self.configs.append(Config(g, face, edges)) + + # per config: list of (coloring, canonical down-apex sequence) + self.colorings: list[list[tuple[Coloring, tuple[int, ...]]]] = [] + # per config: set of canonical sequences it realises + self.config_seq_sets: list[frozenset[tuple[int, ...]]] = [] + # canonical sequence -> list of (config_idx, coloring) + 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, ...]]] = [] + 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)) + self.colorings.append(entries) + self.config_seq_sets.append(frozenset(seqs)) + + def groups(self): + groups: dict[frozenset[tuple[int, ...]], list[int]] = defaultdict(list) + for cidx, sset in enumerate(self.config_seq_sets): + groups[sset].append(cidx) + return sorted(groups.items(), key=lambda kv: (-len(kv[1]), len(kv[0]))) + + def sequences(self) -> list[tuple[int, ...]]: + return sorted(self.by_sequence) + + +# --------------------------------------------------------------------------- +# Drawing (rings the singleton down apexes of the chosen inner face). +# --------------------------------------------------------------------------- + +def _draw(ax, cfg: Config, coloring, title): + graph = cfg.graph + pos = _positions(graph) + for u, v in graph.edges(): + ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]], + color="#bbbbbb", lw=0.5, zorder=1) + for k in range(graph.n): + a, b = f"a{k}", f"a{(k + 1) % graph.n}" + ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]], + color="#666666", lw=1.0, zorder=2) + for v, (x, y) in pos.items(): + is_bite = v.startswith("p") + ax.scatter([x], [y], s=34 if is_bite else 24, color=PALETTE[coloring[v]], + edgecolors="black", linewidths=0.5 if is_bite else 0.3, zorder=3) + # ring the singleton down apexes of this inner face + dx = [pos[v][0] for v in cfg.apexes] + dy = [pos[v][1] for v in cfg.apexes] + ax.scatter(dx, dy, s=120, facecolors="none", edgecolors="#222222", + linewidths=1.4, zorder=4) + ax.set_xlim(-1.65, 1.65) + ax.set_ylim(-1.85, 1.65) + ax.set_aspect("equal") + ax.axis("off") + ax.set_title(title, fontsize=6, pad=1.5) + + +def draw_sequence(exp: Experiment, seq, out_png, out_pdf): + import matplotlib + matplotlib.use("Agg") + import matplotlib.pyplot as plt + + entries = exp.by_sequence[seq] + cols = 10 + rows = math.ceil(len(entries) / cols) + fig, axes = plt.subplots(rows, cols, figsize=(cols * 1.5, rows * 1.7), squeeze=False) + for idx in range(rows * cols): + ax = axes[idx // cols][idx % cols] + if idx < len(entries): + cidx, coloring = entries[idx] + cfg = exp.configs[cidx] + dseq = seq_str(cfg.sequence(coloring)) + _draw(ax, cfg, coloring, f"{config_id(cidx)} d={dseq}") + else: + ax.axis("off") + fig.suptitle( + f"Kempe-balanced colourings with inner-face singleton-down-apex " + f"sequence {seq_str(seq)} (mod colour permutation)\n" + f"n={exp.n}, m={exp.m} down apexes on the face — {len(entries)} " + f"colourings on {len({c for c, _ in entries})} configs; " + f"black rings mark the face's down apexes", + fontsize=11, y=0.998, + ) + fig.tight_layout(rect=(0, 0, 1, 0.96)) + fig.savefig(out_png, dpi=170) + fig.savefig(out_pdf) + plt.close(fig) + print(f"wrote {out_png}") + + +# --------------------------------------------------------------------------- +# Markdown notes. +# --------------------------------------------------------------------------- + +def write_sequence_note(exp: Experiment, seq, path, fig_name): + s = seq_str(seq) + by_config: dict[int, list[Coloring]] = defaultdict(list) + for cidx, coloring in exp.by_sequence[seq]: + by_config[cidx].append(coloring) + + cm: dict[int, int] = {} + for c in seq: + cm[c] = cm.get(c, 0) + 1 + counts = ", ".join(f"{v}×colour{k}" for k, v in sorted(cm.items())) + + lines = [] + lines.append(f"# Inner-face down-apex sequence `{s}`") + 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**." + ) + 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.") + 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[...]` " + "up-tooth apexes; `D[...]` singleton down apexes `d` and bite " + "apexes `p`. Colours 0/1/2 = " + + ", ".join(f"{c}:{PALETTE_NAME[c]}" for c in (0, 1, 2)) + ".") + lines.append("") + for cidx in sorted(by_config): + cfg = exp.configs[cidx] + cols = by_config[cidx] + lines.append(f"## {config_id(cidx)} — {describe_config(cfg.graph, cfg.face, cfg.edges)}") + lines.append("") + lines.append(f"{len(cols)} colouring(s) with down-apex sequence `{s}`:") + lines.append("") + for coloring in cols: + raw = seq_str(cfg.sequence(coloring)) + lines.append(f"- face apexes (raw labels) `{raw}` → canonical `{s}` · " + f"`{compact_coloring(cfg.graph, coloring)}`") + lines.append("") + with open(path, "w") as fh: + fh.write("\n".join(lines) + "\n") + print(f"wrote {path}") + + +def write_summary(exp: Experiment, path): + lines = [] + lines.append(f"# Inner-face singleton-down-apex sequences of Kempe-balanced " + f"colourings (n={exp.n}, m={exp.m})") + lines.append("") + 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)." + ) + lines.append("") + total = sum(len(c) for c in exp.colorings) + lines.append(f"- Total Kempe-balanced colourings (mod colour permutation): " + f"**{total}**.") + lines.append(f"- Distinct canonical down-apex sequences overall: " + f"**{len(exp.by_sequence)}**.") + lines.append("") + lines.append("## Distinct canonical down-apex sequences") + lines.append("") + lines.append("| sequence | colour multiset | #configs realising | #colourings |") + lines.append("|---|---|---|---|") + for seq in exp.sequences(): + cm: dict[int, int] = {} + for c in seq: + cm[c] = cm.get(c, 0) + 1 + cms = "+".join(str(v) for v in sorted(cm.values(), reverse=True)) + ncfg = len({c for c, _ in exp.by_sequence[seq]}) + lines.append(f"| `{seq_str(seq)}` | {cms} | {ncfg} | " + f"{len(exp.by_sequence[seq])} |") + lines.append("") + parity = "even" if exp.m % 2 == 0 else "odd" + allowed = sorted( + {"+".join(str(v) for v in sorted(p, reverse=True)) + for p in _parity_partitions(exp.m)} + ) + lines.append("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 " + f"even number of times). With m = {exp.m} apexes (m is " + f"{'even' if exp.m % 2 == 0 else 'odd'}) every count must be " + f"**{parity}**, so the only admissible colour multisets are " + + ", ".join(allowed) + ".") + lines.append("") + lines.append("## Step 4 — grouping configs by their set of unique down-apex " + "sequences") + lines.append("") + groups = exp.groups() + lines.append(f"The {len(exp.configs)} configs fall into **{len(groups)}** " + f"groups by the set of canonical down-apex sequences they " + f"realise:") + lines.append("") + lines.append("| #configs | set of down-apex sequences | config ids |") + lines.append("|---|---|---|") + for sset, cidxs in groups: + seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset)) + ids = ", ".join(config_id(i) for i in cidxs) + lines.append(f"| {len(cidxs)} | {{ {seqs} }} | {ids} |") + lines.append("") + lines.append("## Config atlas (ids)") + lines.append("") + lines.append("| id | word / bites / face / apexes | #Kempe-balanced | " + "down-apex sequence set |") + lines.append("|---|---|---|---|") + for cidx, cfg in enumerate(exp.configs): + sset = exp.config_seq_sets[cidx] + seqs = ", ".join(f"`{seq_str(s)}`" for s in sorted(sset)) + lines.append(f"| {config_id(cidx)} | " + f"{describe_config(cfg.graph, cfg.face, cfg.edges)} | " + f"{len(exp.colorings[cidx])} | {{ {seqs} }} |") + lines.append("") + lines.append("## Per-sequence notes") + lines.append("") + for seq in exp.sequences(): + lines.append(f"- [`{seq_str(seq)}`](seq_{seq_str(seq)}.md) — " + f"figure `seq_{seq_str(seq)}.png`") + lines.append("") + with open(path, "w") as fh: + fh.write("\n".join(lines) + "\n") + print(f"wrote {path}") + + +# --------------------------------------------------------------------------- +# Driver. +# --------------------------------------------------------------------------- + +def run(args): + exp = Experiment(args.n, args.m) + 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(): + nc = len({c for c, _ in exp.by_sequence[seq]}) + print(f" {seq_str(seq)}: {nc} configs, {len(exp.by_sequence[seq])} colourings") + print(f"groups by sequence-set: {len(exp.groups())}") + + notes_dir = os.path.join(HERE, f"kempe_down_face_sequences_n{args.n}_m{args.m}") + os.makedirs(notes_dir, exist_ok=True) + + write_summary(exp, os.path.join(notes_dir, "summary.md")) + for seq in exp.sequences(): + s = seq_str(seq) + fig_name = f"seq_{s}.png" + write_sequence_note(exp, seq, os.path.join(notes_dir, f"seq_{s}.md"), fig_name) + if not args.no_figures: + draw_sequence(exp, seq, + os.path.join(notes_dir, fig_name), + os.path.join(notes_dir, f"seq_{s}.pdf")) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--n", type=int, default=9) + 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") + run(parser.parse_args()) + + +if __name__ == "__main__": + main() 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 new file mode 100644 index 0000000..d019b39 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_down_face_sequences_n9_m3/seq_012.md @@ -0,0 +1,382 @@ +# 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**. + +- 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. +- 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. + +## C00 — word=UUUUUUDDD bites=- face=root apexes=[d6,d7,d8] + +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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + +## 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]` + 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 new file mode 100644 index 0000000000000000000000000000000000000000..e51f2f5f6d3050400713fe3bbf4390c4455d5d0d GIT binary patch literal 714221 zcmb5W1z40_*EUQD2#6>mNTYOjw^B3oFbdLL(jg^+h;+Af4KOGn-5?+!NJw{gDk=T% 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