Local Morphology of the Partition Graph

TL;DR

This paper investigates the local combinatorial structure of the partition graph $G_n$, describing adjacency and clique properties through bipartite and line graphs based on partition block structures.

Contribution

It provides explicit formulas for degrees, clique classifications, and simplex dimensions in $G_n$, based on a new combinatorial framework involving bipartite graphs.

Findings

The neighborhood of a partition in $G_n$ is isomorphic to a line graph of a bipartite graph.

Explicit formulas for degrees and clique counts are derived.

Local invariants depend only on the support structure of the partition.

Abstract

For a fixed integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with adjacency defined by a single elementary transfer of a cell in the Ferrers diagram. In a previous paper, the clique complex $K_n = \mathrm{Cl}(G_n)$ was studied from a global homotopy-theoretic point of view. This paper studies instead the local combinatorics of the graph $G_n$ itself. For a partition $\lambda=(s_1^{m_1},\dots,s_t^{m_t})$, where $s_1>\dots>s_t>0$, we describe the admissible transfers from $\lambda$ in terms of its block structure. This yields a bipartite graph $B(\lambda)$ obtained from $K_{t,t+1}$ by deleting two explicitly determined families of edges, corresponding to singleton support blocks and unit support gaps. We prove that the graph induced on the neighborhood of $\lambda$ in $G_n$ is isomorphic to the line graph $L(B(\lambda))$. As consequences, we obtain an explicit formula for the degree of $\lambda$, a classification of all cliques through $\lambda$, and a formula for the maximal dimension of a simplex of $K_n$ containing $\lambda$. These local invariants are shown to depend only on an ordered binary datum associated with the support of $\lambda$. The results provide a local structural description of the partition graph and a combinatorial language for the study of larger-scale features of $G_n$.