The Catalan case of Armstrong's conjecture on simultaneous core partitions

A beautiful recent conjecture of D. 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 $\binom{s+1}{3}/2$.