On hook length biases in $t$-regular partitions
Rupam Barman, Pankaj Jyoti Mahanta, Gurinder Singh
TL;DR
The paper investigates hook-length biases in $t$-regular partitions by focusing on the case $t=3$ and the hook length $k=2$. It introduces a combinatorial framework: an injective block-based map $ ext{Φ}$ from $\\\mathcal{B}_3(n)$ to $\\\mathcal{B}_4(n)$ to compare $2$-hook counts, and a partition of the loss set into $\\mathcal{C}_3^1(n)$ and $\\mathcal{C}_3^2(n)$ for targeted compensation. Compensations are achieved via constructing a set $\\mathcal{Q}$ and a compensating map $ ext{Ψ}$, with generating functions $\\mathfrak{X}_3(q)$ and $\\mathfrak{X}_4(q)$ ensuring nonnegative differences, and an induction on $n$ establishes $b_{4,2}(n)\,\ge b_{3,2}(n)$. The Appendix provides detailed case analyses, while the concluding remarks discuss extending the approach to all $t\ge 4$ using a generalized modulo-block scheme, highlighting both the potential and the current limits of the method.
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 $n$. Recently, the first and the third authors proved that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n\geq 4$, and conjectured that $b_{t+1,2}(n)\geq b_{t,2}(n)$ for all $t\geq 3$ and $n\geq 0$. In this paper, we prove that the conjecture is true for $t=3$.