Distributions of Hook Lengths Divisible by Two or Three

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 $t \geq 4$.