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arXiv · 2501.10916

Inequalities and asymptotics for hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions

Eunmi Kim
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Abstract

In this article, we study hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions. More precisely, we establish hook length inequalities between $\ell$-regular partitions and $\ell$-distinct partitions for hook lengths $2$ and $3$, by deriving asymptotic formulas for the total number of hooks of length $t$ in both partition classes, for $t = 1, 2, 3$. From these asymptotics, we show that the ratio of the total number of hooks of length $t$ in $\ell$-regular partitions to those in $\ell$-distinct partitions tends to a constant that depends on $\ell$ and $t$. We also provide hook length inequalities within $\ell$-regular partitions and within $\ell$-distinct partitions.

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Eunmi Kim. 2025-04-13. Inequalities and asymptotics for hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions. https://arxiv.org/abs/2501.10916

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