Hook length biases in ordinary and t-regular partitions

• Published in: Journal of number theory Vol. 264; pp. 41 - 58
• Main Authors: Singh, Gurinder, Barman, Rupam
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.11.2024
• Subjects: Hook lengths; Partition inequalities; Regular partitions; Hook lengths; 11P82; Partition inequalities; Regular partitions; 05A19; 05A15; 05A17
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/j.jnt.2024.05.001

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).