{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T08:48:15Z","timestamp":1768294095502,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Fix coprime $s,t\\ge1$. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the finitely many simultaneous $(s,t)$-cores have average size $\\frac{1}{24}(s-1)(t-1)(s+t+1)$, and that the subset of self-conjugate cores has the same average (first shown by Chen-Huang-Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse stabilizer - giving the \"expected size of the $t$-core of a random $s$-core\" - is $\\frac{1}{24}(s-1)(t^2-1)$. We also prove Fayers' conjecture that the analogous self-conjugate average is the same if $t$ is odd, but instead $\\frac{1}{24}(s-1)(t^2+2)$ if $t$ is even. In principle, our explicit methods - or implicit variants thereof - extend to averages of arbitrary powers.The main new observation is that the stabilizers appearing in Fayers' conjectures have simple formulas in Johnson's $z$-coordinates parameterization of $(s,t)$-cores.We also observe that the $z$-coordinates extend to parameterize general $t$-cores. As an example application with $t := s+d$, we count the number of $(s,s+d,s+2d)$-cores for coprime $s,d\\ge1$, verifying a recent conjecture of Amdeberhan and Leven.\u00a0<\/jats:p>","DOI":"10.37236\/5473","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:13:13Z","timestamp":1578694393000},"source":"Crossref","is-referenced-by-count":17,"title":["Simultaneous Core Partitions: Parameterizations and Sums"],"prefix":"10.37236","volume":"23","author":[{"given":"Victor Y.","family":"Wang","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,1,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i1p4\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i1p4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:37:17Z","timestamp":1579239437000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i1p4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,11]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,1,11]]}},"URL":"https:\/\/doi.org\/10.37236\/5473","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1,11]]},"article-number":"P1.4"}}