Distributions of Hook lengths in integer partitions

Michael Griffin, Ken Ono, Wei-Lun Tsai

Published 2022-01-17, updated 2022-05-19Version 2

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $\mu_t(n)\sim \frac{\sqrt{6n}}{\pi}-\frac{t}{2}$ and variance $\sigma_t^2(n)\sim \frac{(\pi^2-6)\sqrt{6n}}{2\pi^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\geq 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)/2$ and scale $\theta=\sqrt{2/(t-1)}.$