Simplex Stratification and Phase Boundaries in the Partition Graph

TL;DR

This paper introduces a new stratification of the partition graph based on local simplex dimensions, explores phase boundaries between layers, and provides computational insights for small n, revealing patterns and open problems.

Contribution

It formalizes the simplex stratification and phase boundaries in the partition graph, linking local simplex dimensions to star/top capacities and offering computational analysis up to n=30.

Findings

Explicit criteria for layer membership based on star/top capacities.

Identification of boundary thresholds and layer profiles.

Observation of threshold patterns related to staircase partitions.

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 $\lambda$, let $\dim_{\mathrm{loc}}(\lambda)$ denote the largest dimension of a simplex of the clique complex $K_n = \mathrm{Cl}(G_n)$ containing $\lambda$. This defines a decomposition of $V(G_n)$ into layers $L_r(n)=\{\lambda\in V(G_n): \dim_{\mathrm{loc}}(\lambda)=r\}$. 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 $\dim_{\mathrm{loc}}(\lambda)$ is determined exactly by the maximal star and top capacities through $\lambda$. 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.