Large $p$-core $p'$-partitions and walks on the additive residue graph

TL;DR

This paper explores the structure of $p$-core $p'$-partitions, linking their largest size to the longest walks on a specific additive residue graph, and provides bounds matching previous upper bounds.

Contribution

It establishes a connection between $p$-core $p'$-partitions and walks on an additive residue graph, offering explicit constructions and bounds.

Findings

Largest $p$-core $p'$-partition corresponds to longest walk on the residue graph

Explicit family of large $p$-core $p'$-partitions constructed

Lower bound matches previous upper bound by McSpirit and Ono

Abstract

This paper investigates partitions which have neither parts nor hook lengths divisible by $p$, referred to as $p$-core $p'$-partitions. We show that the largest $p$-core $p'$-partition corresponds to the longest walk on a graph with vertices $\{0, 1, \ldots, p-1\}$ and labelled edges defined via addition modulo $p$. We also exhibit an explicit family of large $p$-core $p'$-partitions, giving a lower bound on the size of the largest such partition which is of the same degree as the upper bound found by McSpirit and Ono.