The size of t-cores and hook lengths of random cells in random partitions

Abstract

Fix $\mathit{t}\ge 2$. We first give an asymptotic formula for certain sums of the number of t-cores. We then use this result to compute the distribution of the size of the t-core of a uniformly random partition of an integer n. We show that this converges weakly to a gamma distribution after dividing by $\sqrt{\mathit{n}}$. As a consequence, we find that the size of the t-core is of the order of $\sqrt{\mathit{n}}$ in expectation. We then apply this result to show that the probability that t divides the hook length of a uniformly random cell in a uniformly random partition equals $1/\mathit{t}$ in the limit. Finally, we extend this result to all modulo classes of t using abacus representations for cores and quotients.