A proof of a conjecture of Singh and Barman on hook length

Abstract

Let $b_{t,k}\left(n\right)$ denote the number of hooks of length k in the t-regular partitions of n. In this paper, we focus on inequalities on hook lengths, which Andrews, Ono, Singh and Barman etc. have studied previously. Applying inequalities on the modified Bessel function of the first kind and a modification of the circle method we prove that$b_{3,2}\left(n\right)\ge b_{3,1}\left(n\right)$ holds for $n\ge 28$. This conforms a recent conjecture of Singh and Barman [17]. In addition, we also prove that$b_{2,4}\left(n\right)\ge b_{2,3}\left(n\right)$ holds for $n\ge 82$.