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(k)(n) denote the number of hooks of length k in all the partitions of n. We prove that p(k)(n)≥p(k+1)(n) for all n≥0 and n≠k+1; and p(k)(k+1)−p(k+1)(k+1)=−1 for k≥2. For integers t≥2 and k≥1, let bt,k(n) denote the number of hooks of length k in all the t-regular partitions of n. We find generating functions of bt,k(n) for certain values of t and k. Exploring hook length biases for bt,k(n), we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that b2,2(n)≥b2,1(n) for all n>4, whereas b2,2(n)≥b2,3(n) for all n≥0. We also propose some conjectures on biases among bt,k(n).