$(s,t)$-cores: a weighted version of Armstrong's conjecture

TL;DR

This paper extends Armstrong's conjecture on the average size of $(s,t)$-cores by introducing a weighted version based on group actions, providing new insights into the structure of core partitions.

Contribution

The paper develops a weighted variant of Armstrong's conjecture using affine symmetric group actions, offering a novel perspective on the average sizes of $(s,t)$-cores.

Findings

Weighted average size of $(s,t)$-cores derived

Group actions reveal new structural properties

Expected size of the $t$-core of a random $s$-core

Abstract

The study of core partitions has been very active in recent years, with the study of $(s,t)$-cores - partitions which are both $s$- and $t$-cores - playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive integers, is $\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the same formula gives the average size of a self-conjugate $(s,t)$-core; this was proved by Chen, Huang and Wang. In the present paper, we develop the ideas from the author's paper [J. Combin. Theory Ser. A 118 (2011) 1525-1539] studying actions of affine symmetric groups on the set of $s$-cores in order to give variants of Armstrong's conjectures in which each $(s,t)$-core is weighted by the reciprocal of the order of its stabiliser under a certain group action. Informally, this weighted average gives the expected size of the $t$-core of a random $s$-core.