Hook length biases for self-conjugate partitions and partitions with distinct odd parts

TL;DR

This paper compares hook length distributions between self-conjugate partitions and partitions with distinct odd parts, establishing a bias and asymptotic formulas that resolve a recent conjecture.

Contribution

It introduces the first asymptotic formulas for hook counts in these partition classes and proves a bias favoring self-conjugate partitions for fixed hook lengths.

Findings

More hooks of fixed length in self-conjugate partitions for large n

Asymptotic formulas for hook counts in both classes

Resolution of a conjecture by Ballantine et al.

Abstract

We establish a hook length bias between self-conjugate partitions and partitions of distinct odd parts, demonstrating that there are more hooks of fixed length $t \geq 2$ among self-conjugate partitions of $n$ than among partitions of distinct odd parts of $n$ for sufficiently large $n$. More precisely, we derive asymptotic formulas for the total number of hooks of fixed length $t$ in both classes. This resolves a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.