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

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 (s+1 over 3)/2