When Does the Set of $(a, b, c)$-Core Partitions Have a Unique Maximal Element?

TL;DR

This paper characterizes when a triple of positive integers $(a, b, c)$ admits a maximal $(a, b, c)$-core partition containing all others, extending known results from pairs to triples in the context of core partitions.

Contribution

It provides a complete characterization for the existence of a maximal $(a, b, c)$-core partition when $a$, $b$, and $c$ are pairwise coprime, generalizing previous pairwise results.

Findings

Established necessary and sufficient conditions for the existence of a maximal $(a, b, c)$-core.

Extended Olsson and Stanton's result from pairs to triples of integers.

Provided a comprehensive answer to Fayers' question for pairwise coprime triples.

Abstract

In 2007, Olsson and Stanton gave an explicit form for the largest $(a, b)$-core partition, for any relatively prime positive integers $a$ and $b$, and asked whether there exists an $(a, b)$-core that contains all other $(a, b)$-cores as subpartitions; this question was answered in the affirmative first by Vandehey and later by Fayers independently. In this paper we investigate a generalization of this question, which was originally posed by Fayers: for what triples of positive integers $(a, b, c)$ does there exist an $(a, b, c)$-core that contains all other $(a, b, c)$-cores as subpartitions? We completely answer this question when $a$, $b$, and $c$ are pairwise relatively prime; we then use this to generalize the result of Olsson and Stanton.