{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:21:30Z","timestamp":1759335690158,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The study of core partitions has been very active in recent years, with the study of $(s,t)$-cores \u2014 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\u20141539], 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.<\/jats:p>","DOI":"10.37236\/6161","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:50:43Z","timestamp":1578685843000},"source":"Crossref","is-referenced-by-count":4,"title":["$(s,t)$-Cores: a Weighted Version of Armstrong\u2019s Conjecture"],"prefix":"10.37236","volume":"23","author":[{"given":"Matthew","family":"Fayers","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,11,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p32\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:10:09Z","timestamp":1579237809000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i4p32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,25]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6161","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,11,25]]},"article-number":"P4.32"}}