Hook length biases in ordinary and t-regular partitions

2024/04/11 by Gurinder Singh, Singh, Gurinder, Rupam Barman +1 · 1 citation
Mathematics · Computer Science · #Advanced Mathematical Identities #Coding theory and cryptography #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2404.07485

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 b_t,k(n) denote the number of hooks of length k in all the t-regular partitions of n. We find generating functions of b_t,k(n) for certain values of t and k. Exploring hook length biases for b_t,k(n), we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that b_2,2(n)≥ b_2,1(n) for all n>4, whereas b_2,2(n)≥ b_2,3(n) for all n≥ 0. We also propose some conjectures on biases among b_t,k(n).

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• Proof of Singh and Barman's conjecture on hook length biases

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