Simultaneous Core Partitions: Parameterizations and Sums

Abstract

Fix coprime s; t > 1. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the definitely many simultaneous (s; t)- cores have average size 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 1 24 (s - 1)(t2 - 1). We also prove Fayers' conjecture that the analogous self-conjugate average is the same if t is odd, but instead 1 24 (s - 1)(t2 + 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 > 1, verifying a recent conjecture of Amdeberhan and Leven.