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

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.