Average Size of a Self-conjugate (s, t)-Core Partition

TL;DR

This paper proves a conjecture that the average size of self-conjugate (s,t)-core partitions equals a specific formula, using a bijection with lattice paths, extending previous results on core partitions.

Contribution

It establishes the average size of self-conjugate (s,t)-core partitions, confirming a conjecture through a novel bijection with lattice paths.

Findings

Average size matches the conjectured formula

Bijection with lattice paths provides a new proof method

Extends known results to self-conjugate core partitions

Abstract

Armstrong, Hanusa and Jones conjectured that if $s,t$ are coprime integers, then the average size of an $(s,t)$-core partition and the average size of a self-conjugate $(s,t)$-core partition are both equal to $\frac{(s+t+1)(s-1)(t-1)}{24}$. Stanley and Zanello showed that the average size of an $(s,s+1)$-core partition equals $\binom{s+1}{3}/2$. Based on a bijection of Ford, Mai and Sze between self-conjugate $(s,t)$-core partitions and lattice paths in $\lfloor \frac{s}{2} \rfloor\times \lfloor \frac{t}{2}\rfloor$ rectangle, we obtain the average size of a self-conjugate $(s,t)$-core partition as conjectured by Armstrong, Hanusa and Jones.