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Hook length inequalities for t-regular partitions in the t-aspect
Published online by Cambridge University Press: 24 July 2025
Abstract
Let 
$t\geq 2$ and 
$k\geq 1$ be integers. A t-regular partition of a positive integer n is a partition of n such that none of its parts is divisible by t. Let 
$b_{t,k}(n)$ denote the number of hooks of length k in all the t-regular partitions of n. In this article, we prove some inequalities for 
$b_{t,k}(n)$ for fixed values of k. We prove that for any 
$t\geq 2$, 
$b_{t+1,1}(n)\geq b_{t,1}(n)$, for all 
$n\geq 0$. We also prove that 
$b_{3,2}(n)\geq b_{2,2}(n)$ for all 
$n>3$, and 
$b_{3,3}(n)\geq b_{2,3}(n)$ for all 
$n\geq 0$. Finally, we state some problems for future works.
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society
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