Hook length biases in ordinary and t-regular partitions

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. Our proofs are both combinatorial and analytic in nature. We also propose some conjectures on biases among bt,k(n).