Hook length biases in ordinary and t-regular partitions

Abstract

In this article, we study hook lengths of ordinary partitions and t-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in 2-regular partitions. For a positive integer k, let $p_{\left(k\right)}\left(n\right)$ denote the number of hooks of length k in all the partitions of n. We prove that $p_{\left(k\right)}\left(n\right)\ge p_{(k+1)}\left(n\right)$ for all $n\ge 0$ and $n\ne k+1$; and $p_{\left(k\right)}(k+1)-p_{(k+1)}(k+1)=-1$ for $k\ge 2$. For integers $t\ge 2$ and $k\ge 1$, let $b_{t,k}\left(n\right)$ denote the number of hooks of length k in all the t-regular partitions of n. We find generating functions of $b_{t,k}\left(n\right)$ for certain values of t and k. Exploring hook length biases for $b_{t,k}\left(n\right)$, we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that $b_{2,2}\left(n\right)\ge b_{2,1}\left(n\right)$ for all $n>4$, whereas $b_{2,2}\left(n\right)\ge b_{2,3}\left(n\right)$ for all $n\ge 0$. We also propose some conjectures on biases among $b_{t,k}\left(n\right)$.