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

TL;DR

This paper investigates hook length inequalities and asymptotic behaviors in $ ext{ell}$-regular and $ ext{ell}$-distinct partitions, revealing ratios and bounds for hook counts of lengths 1, 2, and 3.

Contribution

It establishes new inequalities and asymptotic formulas for hook lengths in $ ext{ell}$-regular and $ ext{ell}$-distinct partitions, including ratio limits depending on $ ext{ell}$ and hook length.

Findings

Ratios of hook counts tend to constants depending on $ ext{ell}$ and $t$

Hook length inequalities are established within each partition class

Asymptotic formulas for total hooks of length $t$ in both classes

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.