Simultaneous Core Partitions: Parameterizations and Sums

Fix coprime s, t > 1. 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 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)(t 2 − 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)(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 > 1, verifying a recent conjecture of Amdeberhan and Leven.