Distributions of Hook Lengths Divisible by Two or Three

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 $t\ge 4$.