Hook-Length Biases in $t$-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.