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

TL;DR

This work analyzes hook lengths in $\ell$-regular and $\ell$-distinct partitions, establishing generating-function frameworks and circle-method–driven asymptotics for the counts of hooks of lengths $t=1,2,3$ in each class. It proves that the ratios $\frac{d_{\ell,t}(n)}{b_{\ell,t}(n)}$ converge to positive constants $r_{\ell,t}$, with $r_{\ell,t}\to1$ as $\ell\to\infty$, and shows intra- and inter-class inequalities that hold for large $n$. The results rely on explicit generating functions $B_{\ell,t}(q)$ and $D_{\ell,t}(q)$, modular transformations, Bernoulli/digamma machinery, Euler-Maclaurin summation, and Bessel-function estimates to obtain precise asymptotics. Together, these findings provide a unified asymptotic picture of hook-length distributions across $\ell$-regular and $\ell$-distinct partitions and quantify how the two families compare as $n$ grows large.

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.

Figures (4)

• Figure 1: Arm, coarm, and leg lengths of $v$: $a_\lambda(v)=j$, $ca_\lambda(v)=m$, and $l_\lambda(v)=i$
• Figure 2: The Young diagram of the partition ${(5,4, 2, 1)}$ with hook lengths
• Figure 3: Cases for a hook of length $2$
• Figure 4: Cases for a hook of length $3$

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Remark
• Corollary 1.6
• Lemma 2.1: BJM
• Lemma 2.2: BB
• Theorem 3.1