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

Matthew Fayers

The electronic journal of combinatorics, Tome 23 (2016) no. 4

Résumé

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 %mdash; 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.

Détail

DOI : 10.37236/6161

Classification : 05A17, 05A15
Mots-clés : partition, core, affine symmetric group, affine hyperoctahedral group, Armstrong's conjecture