  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 $\binom{s+1}{3}/2$.
