On the hook length biases of the $2$- and $3$-regular partitions

TL;DR

This paper investigates hook length biases in 2- and 3-regular partitions, confirming one conjecture, disproving another for infinitely many cases, and verifying it for specific values, while proposing a new conjecture for even cases.

Contribution

It confirms one conjecture on hook length differences, disproves another for infinitely many odd cases, verifies it for certain even cases, and proposes a new related conjecture.

Findings

Confirmed positivity conjecture for 3,2 hooks.

Disproved the second conjecture for infinitely many odd k.

Verified the second conjecture for k=4 and 6.

Abstract

Let $b_{t,i}(n)$ denote the total number of the $i$ hooks in the $t$-regular partitions of $n$. Singh and Barman (J. Number Theory { 264} (2024), 41--58) raised two conjectures on $b_{t,i}(n)$. The first conjecture is on the positivity of $b_{3,2}(n)-b_{3,1}(n)$ for $n\ge 28$. The second conjecture states that when $k\ge 3$, $b_{2,k}(n)\ge b_{2,k+1}(n)$ for all $n$ except for $n= k+1$. In this paper, we confirm the first conjecture. {Moreover, we show that for any odd $k\ge 3$, the second conjecture fails for infinitely many $n$.} {Furthermore, we verify that the second conjecture holds for $k=4$ and $6$.} We also propose a conjecture on the even case $k$, which is a modification of Singh and Barman's second conjecture.