Add small-n ELG enumeration table to even_level_graph_generators

Records, for 4<=n<=11, triangulation iso classes, how many admit an ELG
source, ELG iso classes, and the automorphism-free flag-rooted count
sum_G 4E/|Aut(G)| * s(G). Computed by experiments/count_elgs.py.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

papers/even_level_graph_generators/paper.tex

@@ -310,6 +310,51 @@ $\{\text{yellow}, \text{green}\}$. The combined assignment is a
 proper $4$-coloring of $G$.
 \end{proof}
 
+\subsection*{Enumeration for small $n$}
+
+Even Level Graphs are scarce among triangulations. For each $n$ we
+enumerated all iso classes of plane triangulations and tested, for every
+choice of source vertex, whether all level subgraphs are bipartite
+(equivalently, whether every level cycle is even).
+Table~\ref{tab:elg-counts} records, for $4 \leq n \leq 11$: the number of
+triangulation iso classes; how many of them admit at least one Even Level
+Graph source; the number of Even Level Graph iso classes (pairs $(G, S)$
+up to isomorphism, i.e.\ valid sources counted up to $\mathrm{Aut}(G)$);
+and the number of \emph{flag-rooted} Even Level Graphs,
+\[
+  \sum_{G} \frac{4E}{|\mathrm{Aut}(G)|}\, s(G),
+  \qquad E = 3n - 6,
+\]
+where $s(G)$ is the number of valid sources of $G$. The flag-rooting is the
+automorphism-free count: $\mathrm{Aut}(G)$ acts freely on the $4E$ flags of a
+$3$-connected triangulation, so every summand is an integer.
+
+The smallest Even Level Graph is the octahedron at $n = 6$: from any vertex
+the four neighbours form a $4$-cycle at level $1$ and the antipode sits alone
+at level $2$. Below $n = 6$ every triangulation forces an odd level cycle, so
+no Even Level Graph exists.
+
+\begin{table}[ht]
+\centering
+\begin{tabular}{ccccc}
+$n$ & triangulations & with ELG source & ELG iso classes & flag-rooted ELGs \\\hline
+$4$  & $1$    & $0$   & $0$   & $0$ \\
+$5$  & $1$    & $0$   & $0$   & $0$ \\
+$6$  & $2$    & $1$   & $1$   & $6$ \\
+$7$  & $5$    & $2$   & $2$   & $45$ \\
+$8$  & $14$   & $5$   & $6$   & $186$ \\
+$9$  & $50$   & $13$  & $14$  & $651$ \\
+$10$ & $233$  & $37$  & $45$  & $2766$ \\
+$11$ & $1249$ & $129$ & $169$ & $14346$ \\
+\end{tabular}
+\caption{Even Level Graph counts for $4 \leq n \leq 11$. The
+\emph{triangulations} column is the number of plane-triangulation iso classes
+(OEIS A000109). \emph{ELG iso classes} counts pairs $(G, S)$ up to
+isomorphism; \emph{flag-rooted ELGs} is the automorphism-free count
+$\sum_G \tfrac{4E}{|\mathrm{Aut}(G)|}\,s(G)$.}
+\label{tab:elg-counts}
+\end{table}
+
 \begin{definition}[Derived level graph]
 \label{def:derived-level-graph}
 Let $G$ be an Even Level Graph with level source $S$, and let $E$ and