Simultaneous core partitions: parameterizations and sums

TL;DR

This paper provides new proofs and formulas for the average sizes of simultaneous core partitions, including self-conjugate cases, using Johnson's $z$-coordinates, and extends these methods to count specific multi-core partitions.

Contribution

It offers new proofs of existing conjectures on core partition averages without Ehrhart reciprocity and introduces a simple formula for stabilizers in Johnson's coordinates.

Findings

Average size of (s,t)-cores is (s-1)(t-1)(s+t+1)/24

Expected size of the t-core of a random s-core is (s-1)(t^2-1)/24

Count of (s,s+d,s+2d)-cores matches recent conjecture

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.