The inequality on the number of 1-hooks, 2-hooks and 3-hooks in t-regular partitions
Abstract
Let bn,k denote the number of hooks of length k in all the t-regular partitions of n. Singh and Barman raised the question of finding the relation between bt,2(n) and bt,1(n). Kim showed that there exists N such that bt,2(n) bt,1(n) and bt,2(n) ≥ bt,3(n) for n>N. In this paper, we find an explicit bound of N=O(t5) for bt,2(n)≥ bt,1(n) and show that bt,2(n) ≥ bt,3(n) for all n 4.
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