Local Morphology of the Partition Graph

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 $λ=(s_1^{m_1},\dots,s_t^{m_t})$, where $s_1>\dots>s_t>0$, we describe the admissible transfers from $λ$ in terms of its block structure. This yields a bipartite graph $B(λ)$ 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 $λ$ in $G_n$ is isomorphic to the line graph $L(B(λ))$. As consequences, we obtain an explicit formula for the degree of $λ$, a classification of all cliques through $λ$, and a formula for the maximal dimension of a simplex of $K_n$ containing $λ$. These local invariants are shown to depend only on an ordered binary datum associated with the support of $λ$. The results provide a local structural description of the partition graph and a combinatorial language for the study of larger-scale features of $G_n$.