Distributions of Hook Lengths Divisible by Two or Three

TL;DR

This paper studies the distribution of hook lengths divisible by 2 or 3 in partitions of integers, showing that their cumulative distributions converge to shifted Gamma distributions as n grows large.

Contribution

It characterizes the support and asymptotic behavior of the distributions of hook lengths divisible by 2 or 3, extending previous work for larger divisors.

Findings

Support of distributions is small for large n

Mass functions approximate continuous functions

CDFs converge to shifted Gamma distributions

Abstract

For fixed $t = 2$ or $3$, we investigate the statistical properties of $\{Y_t(n)\}$, the sequence of random variables corresponding to the number of hook lengths divisible by $t$ among the partitions of $n$. We characterize the support of $Y_t(n)$ and show, in accordance with empirical observations, that the support is vanishingly small for large $n$. Moreover, we demonstrate that the nonzero values of the mass functions of $Y_2(n)$ and $Y_3(n)$ approximate continuous functions. Finally, we prove that although the mass functions fail to converge, the cumulative distribution functions of $\{Y_2(n)\}$ and $\{Y_3(n)\}$ converge pointwise to shifted Gamma distributions, completing a characterization initiated by Griffin--Ono--Tsai for $t \geq 4$.