Simultaneous core partitions: parameterizations and sums

Abstract

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.