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

Abstract

Fix $t \geq 2$. 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 $\sqrt{n}$. As a consequence, we find that the size of the $t$-core is of the order of $\sqrt{n}$ 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 $1/t$ in the limit. Finally, we extend this result to all modulo classes of $t$ using abacus representations for cores and quotients.Comment: 28 pages, 3 figures, significant revisions. Several minor errors fixed and results stated in a more concise manner. From v1, Sections 2.4 and 5.2 deleted and Corollary 5.6 is stated as Lemma 5.16 in this version, thanks to a suggestion of D. Grinber