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Armstrong, Hanusa and Jones conjectured that if $s,t$ are coprime integers,then the average size of an $(s,t)$-core partition and the average size of aself-conjugate $(s,t)$-core partition are both equal to$\frac{(s+t+1)(s-1)(t-1)}{24}$. Stanley and Zanello showed that the averagesize of an $(s,s+1)$-core partition equals $\binom{s+1}{3}/2$. Based on abijection of Ford, Mai and Sze between self-conjugate $(s,t)$-core partitionsand lattice paths in $\lfloor \frac{s}{2} \rfloor\times \lfloor\frac{t}{2}\rfloor$ rectangle, we obtain the average size of a self-conjugate$(s,t)$-core partition as conjectured by Armstrong, Hanusa and Jones.

Average Size of a Self-conjugate (s, t)-Core Partition