Large \(p\)-Core \(p'\)-Partitions and Walks on the Additive Residue Graph

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.