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

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