Hook-Length Biases in $t$-regular partitions

TL;DR

The paper addresses hook-length biases in $t$-regular partitions by deriving purely analytic $q$-series proofs for key inequalities between hook-length counts, notably showing $b_{3,2}(n)\ge b_{2,2}(n)$ for $n>3$ and providing an alternate analytic route to $b_{2,2}(n)\ge b_{2,1}(n)$. Central to the approach is the generating-function framework $B_{t,k}(q)$, with strategic manipulations of $G(q)=B_{3,2}(q)-B_{2,2}(q)$ and its multiplier $F(q)=(1-q^6)G(q)$, and the use of Euler and Sylvester identities alongside monotonicity properties of $d_3(n)$. A key corollary reveals a direct connection between $b_{2,2}(n)-b_{2,1}(n)$ and distinct-parts partitions with parts at least $3$, namely $d_3(n)$, via a precise summation identity, and a combinatorial injection underscores this relationship. Altogether, the work expands the repertoire of techniques to study hook-length statistics in $t$-regular partitions and links these statistics to distinct-part partition structures, enriching the combinatorial and $q$-series landscape.

Abstract

Recently, there has been a lot of work on combinatorial inequalities related to hook-lengths in $t$-regular partitions. In this short note, we give a proof using generating functions for a result proved by Singh and Barman (2026) using combinatorial methods. In addition, we give an alternate proof of another result of Singh \& Barman (2024) which yields as a corollary a previously unobserved connection of hook-lengths in $t$-regular partitions with certain distinct parts partitions.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof : Proof of Lemma \ref{['lem:d3-monotone']}
• Lemma 3.1