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

Abstract

Fix $t\ge 2$. 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 $\sqrt{n}$. 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.

29 pages, 3 figures, minor changes, final version

References in corpus (1)

• Core size of a random partition for the Plancherel measure

Cited by in corpus (3)

• Core size of a random partition for the Plancherel measure
• Asymptotic behavior of Wronskian polynomials that are factorized via $p$-cores and $p$-quotients
• Cores of partitions in rectangles