Hook length inequalities for $t$-regular partitions in the $t$-aspect

TL;DR

This paper investigates inequalities related to hook lengths in $t$-regular partitions, establishing several bounds and comparisons for the number of hooks of fixed lengths across different regularity parameters.

Contribution

It proves new inequalities for the counts of hooks of fixed lengths in $t$-regular partitions, advancing understanding of their combinatorial properties.

Findings

Proved that $b_{t+1,1}(n) \,\geq\, b_{t,1}(n)$ for all $n\geq0$.

Established that $b_{3,2}(n) \,\geq\, b_{2,2}(n)$ for all $n>3$.

Showed that $b_{3,3}(n) \,\geq\, b_{2,3}(n)$ for all $n\geq0$.

Abstract

Let $t\geq2$ and $k\geq1$ 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 $t\geq2$, $b_{t+1,1}(n)\geq b_{t,1}(n)$, for all $n\geq0$. 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 $n\geq0$. Finally, we state some problems for future works.