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

TL;DR

This paper investigates the asymptotic behavior of the size of t-cores in random partitions, showing it converges to a gamma distribution scaled by sqrt(n), and analyzes hook length divisibility probabilities.

Contribution

It provides the first asymptotic formula for sums of t-cores and characterizes the distribution of t-core sizes in random partitions, extending to hook length divisibility probabilities.

Findings

Size of t-core scales with sqrt(n) in expectation

Distribution of t-core size converges to a gamma distribution

Probability that t divides hook length approaches 1/t

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.