Simplex Stratification and Phase Boundaries in the Partition Graph
Fedor B. Lyudogovskiy
Abstract
We study the partition graph $G_n$, whose vertices are the integer partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts. We introduce the simplex stratification of $G_n$: for each vertex $λ$, let $\dim_{\mathrm{loc}}(λ)$ denote the largest dimension of a simplex of the clique complex $K_n = \mathrm{Cl}(G_n)$ containing $λ$. This defines a decomposition of $V(G_n)$ into layers $L_r(n)=\{λ\in V(G_n): \dim_{\mathrm{loc}}(λ)=r\}$. We formalize the graph-theoretic interfaces between consecutive layers, called phase boundaries, and study the associated interface graphs and boundary thresholds. Using the previously established star/top description of cliques through a fixed vertex, we show that $\dim_{\mathrm{loc}}(λ)$ is determined exactly by the maximal star and top capacities through $λ$. This yields explicit local criteria for membership in higher simplex layers and reformulates their first appearance in terms of local star/top capacity thresholds. We also present an exhaustive computational study for $n\le 30$, including exact-layer thresholds, boundary thresholds, selected layer profiles, and the behaviour of the boundary framework. The computations suggest a rigid threshold pattern related to staircase partitions and their one-cell extensions, while the corresponding global statements are left as conjectures and open problems.