Distributions of Hook Lengths Divisible by Two or Three

Hannah Lang; Hamilton Wan; Nancy Xu

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arXiv · 2207.06486

Distributions of Hook Lengths Divisible by Two or Three

Hannah Lang·Hamilton Wan·Nancy Xu

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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 $t \geq 4$.

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Hannah Lang, Hamilton Wan, Nancy Xu. 2022-07-13. Distributions of Hook Lengths Divisible by Two or Three. https://arxiv.org/abs/2207.06486

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