Hook Length Biases in $t$-Core Partitions

TL;DR

This paper extends the study of hook length biases to $t$-core partitions, revealing inequalities in hook counts for various $t$ and $k$, using mainly combinatorial methods.

Contribution

It introduces the analysis of hook length biases specifically for $t$-core partitions, a new area in the ongoing research.

Findings

For $t=3$, $a_{3,1}(n)igge a_{3,2}(n)igge a_{3,4}(n)$

For $t=4$, $a_{4,1}(n)igge a_{4,3}(n)$

The results are established for all $n$ using combinatorial techniques.

Abstract

Recently, the theory of hook length biases has emerged as a prominent research topic. Led by Ballantine, Burson, Craig, Folsom, and Wen [\textit{Res. Math. Sci.}, 2023], hook length biases are being explored for ordinary partitions, odd versus distinct partitions, self-conjugate versus distinct odd partitions. Lately, Singh and Barman [\textit{J. Number Theory}, 2024] opened the door to hook length biases in $\ell$-regular partitions. In this work, we extend the theory of hook length biases to $t$-core partitions. For example, let $a_{t,k}(n)$ denote the number of hooks of length $k$ in all $t$-core partitions of $n$, then we find that $a_{3,1}(n)\ge a_{3,2}(n) \ge a_{3,4}(n)$ and $a_{4,1}(n)\ge a_{4,3}(n)$ for all $n$. The methods employed in this work are mainly combinatorial.