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On Singh and Barman's Conjecture on Hook Length Biases in \\(t\\)-Regular Partitions
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On Singh and Barman's Conjecture on Hook Length Biases in \\(t\\)-Regular Partitions
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Paper
On Singh and Barman's Conjecture on Hook Length Biases in \\(t\\)-Regular Partitions
2024
Overview
Let \\(b_{t,k}(n)\\) denote the number of hooks of length \\(k\\) in all the \\(t\\)-regular partitions of \\(n\\), where \\(t\\geq 2\\) and \\(k\\geq 1\\). Recently, Singh and Barman proved that \\(b_{3,2}(n)\\geq b_{2,2}(n)\\) for all \\(n\\geq 4\\). They also conjectured that \\(b_{t+1,2}(n)\\geq b_{t,2}(n)\\) for \\(t\\geq 3\\) and for all \\(n\\geq 0\\). In this paper, we prove that the conjecture is true for \\(t=3\\).
Publisher
Cornell University Library, arXiv.org
Subject
