Support and Support Jumps in the Partition Graph

TL;DR

This paper investigates the properties of the support of partitions within the partition graph, revealing support size conditions, support jump bounds, and structural invariants, supported by computational data.

Contribution

It provides exact characterizations of support size occurrences, support jumps, and degree bounds, along with structural invariants and computational analysis of the partition graph.

Findings

Support size r occurs iff T_r = r(r+1)/2 ≤ n.

Support jumps are always in {-2,-1,0,1,2}.

Degree bounds are established with equality for staircase partitions.

Abstract

Let $G_n$ be the partition graph whose vertices are the partitions of $n$, with adjacency given by elementary transfers of one cell between parts, followed by reordering. We study the support of a partition -- the set of distinct part sizes -- as a global vertex invariant of $G_n$. We show that support size $r$ occurs in $G_n$ if and only if $T_r=r(r+1)/2\le n$, so the maximal support size is $\rho(n)=\max\{r:T_r\le n\}$. We determine exactly how support changes along an edge: the support jump always lies in $\{-2,-1,0,1,2\}$, and we give an explicit birth-death formula in terms of the source and target part sizes. We also prove the degree bound $\deg(\lambda)\ge \sigma(\lambda)(\sigma(\lambda)-1)$ for every partition $\lambda$, with equality exactly for staircase partitions. In addition, support size is invariant under conjugation, the support-$1$ stratum consists exactly of rectangular partitions, and the coarse support-level graph always contains the chain $1-2-\cdots-\rho(n)$. We conclude with computational data for small $n$, including support-stratum counts, support-jump counts, and connectivity data for fixed-support subgraphs.