Hook length inequalities for <i>t</i> -regular partitions in the <i>t</i> -aspect

Abstract Let $t\geq 2$ and $k\geq 1$ be integers. A t -regular partition of a positive integer n is a partition of n such that none of its parts is divisible by t . Let $b_{t,k}(n)$ denote the number of hooks of length k in all the t -regular partitions of n . In this article, we prove some inequalities for $b_{t,k}(n)$ for fixed values of k . We prove that for any $t\geq 2$ , $b_{t+1,1}(n)\geq b_{t,1}(n)$ , for all $n\geq 0$ . We also prove that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n&gt;3$ , and $b_{3,3}(n)\geq b_{2,3}(n)$ for all $n\geq 0$ . Finally, we state some problems for future works.