The Catalan Case of Armstrong's Conjecture on Simultaneous Core Partitions

Source abstract

A beautiful recent conjecture of Armstrong predicts the average size of a partition that is simultaneously an $s$-core and a $t$-core, where $s$ and $t$ are coprime. Our goal is to prove this conjecture when t=s+1. These simultaneous (s,s+1)-core partitions, which are enumerated by Catalan numbers, have average size $\binom{s+1}{3}/2$.