Simplicial shells and thickness in the partition graph

TL;DR

This paper investigates the local simplicial dimension in the clique complex of a graph formed by partitions of integers, revealing invariant properties and shell structures up to n=30.

Contribution

It introduces the concept of simplicial thickness as a local invariant, analyzes its properties, and provides a finite computational atlas for small n.

Findings

Simplicial thickness is preserved by conjugation.

Thick zones and shells are conjugation-invariant.

The first nontrivial shell appears at order 2, called the triangular skin.

Abstract

For each positive integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with edges given by elementary transfers of one unit between parts, followed by reordering. We study the local simplex dimension in the clique complex $K_n=\Cl(G_n)$ as a geometric thickness invariant of $G_n$. For a partition $\lambda\vdash n$, let $\tau_n(\lambda):=\dim_{\mathrm{loc}}(\lambda)$ be its simplicial thickness. This gives threshold thick zones $T_{\ge r}(n)=\{\lambda: \tau_n(\lambda)\ge r\}$ and, relative to the boundary framework of $G_n$, a shell/core decomposition into outer shells $Sh_r(n)$ and inner cores $Core_r(n)$. Using local-morphology results established earlier in the series, we work with simplicial thickness as a local invariant. We prove that it is preserved by conjugation, that the induced thick zones, shells, and cores are conjugation-invariant, and that the antennas remain strictly one-dimensional in the simplicial sense and are excluded from all nontrivial thick zones. The first shell order at which a nontrivial shell can occur is therefore $2$, and the corresponding shell $Sh_2(n)$ is the triangular skin, while higher simplicial regimes form nested higher-order shells inside the triangular regime. We also develop a complete finite computational atlas for $1\le n\le 30$, giving first-occurrence tables for the regimes $T_{\ge r}(n)$ and supporting a finite-range rear-central thickening pattern.