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arXiv · 2510.19185

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

Hongshu Lin·Wenston J. T. Zang
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Abstract

Let $b_{t,i}(n)$ denote the total number of $i$-hooks in $t$-regular partitions of $n$. Singh and Barman conjectured that $b_{t+1,2}(n) \geq b_{t,2}(n)$ holds for all $t\ge 3$ and $n\ge 0$. This conjecture was known to hold for $t=3$ due to work of Barman Mahanta and Singh. In this paper, we prove this conjecture.

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Hongshu Lin, Wenston J. T. Zang. 2025-11-19. Proof of Singh and Barman's conjecture on hook length biases. https://arxiv.org/abs/2510.19185

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