Hook-Length Biases in $t$-regular partitions

TL;DR

This paper provides generating function proofs for recent results on hook-length inequalities in t-regular partitions, offering alternative approaches and uncovering new connections with partitions having distinct parts.

Contribution

It introduces generating function proofs for existing combinatorial results and reveals a novel link between hook-lengths in t-regular partitions and partitions with distinct parts.

Findings

Generated alternative proofs for known inequalities

Discovered a new connection between hook-lengths and distinct parts partitions

Enhanced understanding of combinatorial properties of t-regular partitions

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.