Simplex Layers and Phase Boundaries in the Partition Graph

Abstract

For the partition graph $G_n$ on the set of partitions of $n$, we study the stratification induced by the local simplex dimension $\dim_{\mathrm{loc}}(\lambda)$, defined as the maximal dimension of a simplex of the clique complex $K_n=\mathrm{Cl}(G_n)$ containing $\lambda$. Using the previously established description of maximal cliques through a vertex in terms of star and top capacities, we define the simplex layers $L_r(n):=\{\lambda\vdash n:\dim_{\mathrm{loc}}(\lambda)=r\}$ and study their global structure. We formalize the resulting layer stratification, rewrite layer membership in terms of local capacities, and record its basic consequences, including conjugation invariance. We then investigate first occurrence of layers across $n$, introducing the indices $n_r^{\mathrm{first}}$ and the corresponding first-occurrence sets $\mathcal{F}_r$. For the initial layer values, we obtain explicit exact results; more generally, we record a finite first-occurrence table and several natural sequence questions. We also define the adjacent-layer edge boundary $\partial^E_{r,r+1}(n)$, consisting of edges joining $L_r(n)$ to $L_{r+1}(n)$, together with the associated one-sided and vertex-boundary variants. This provides an exact interface language for the layer stratification, distinct from the broader shell-type geometric language used elsewhere in the project.