(s, t)-Cores: a Weighted Version of Armstrong's Conjecture

The study of core partitions has been very active in recent years, with the study of ps, tq-cores – partitions which are both sand t-cores – playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an ps, tq-core, when s and t are coprime positive integers, is 1 24ps ́1qpt ́1qps` t ́1q. Armstrong also conjectured that the same formula gives the average size of a selfconjugate ps, tq-core; this was proved by Chen, Huang and Wang. In the present paper, we develop the ideas from the author’s paper [F1], studying actions of affine symmetric groups on the set of s-cores in order to give variants of Armstrong’s conjectures in which each ps, tq-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.