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

Abstract

Let $b_{t,i}\left(n\right)$ denote the total number of i hooks in the t-regular partitions of n. Singh and Barman (2024) [14] raised two conjectures on $b_{t,i}\left(n\right)$. The first conjecture is on the positivity of $b_{3,2}\left(n\right)-b_{3,1}\left(n\right)$ for $n\ge 28$. The second conjecture states that when $k\ge 3$, $b_{2,k}\left(n\right)\ge b_{2,k+1}\left(n\right)$ 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.