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

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.