The Catalan case of Armstrong's conjecture on simultaneous core partitions

A beautiful recent conjecture of D. Armstrong predicts the average size of a partition that is simultaneously an $s$-core and a $t$-core, where $s$ and $t$ are coprime. Our goal is to prove this conjecture when $t=s+1$. These simultaneous $(s,s+1)$-core partitions, which are enumerated by Catalan numbers, have average size $\binom{s+1}{3}/2$.Comment: Some changes in response to the referee's comments. To appear in the SIAM J. on Discrete Mat

The Catalan case of Armstrong\u27s conjecture on simultaneous core partitions Read More: https://epubs.siam.org/doi/10.1137/130950318

A beautiful recent conjecture of D. Armstrong predicts the average size of a partition that is simultaneously an s-core and a t-core, where s and t are coprime. Our goal is to prove this conjecture when t = s + 1. These simultaneous (s, s + 1)-core partitions, which are enumerated by Catalan numbers, have average size ((s+1)/3)/2

(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 $(s,t)$-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