Inequalities and asymptotics for hook numbers in restricted partitions

TL;DR

This paper derives asymptotic formulas for hook numbers in restricted partitions, compares their distributions in odd and distinct partitions, and proves related conjectures and probabilistic properties.

Contribution

It introduces new asymptotic formulas for hook numbers in restricted partitions and confirms a conjecture about their distribution in odd versus distinct partitions.

Findings

Partitions into odd parts have more hooks of size h on average for large n.

Asymptotic formulas for the total number of hooks of size h are established.

Probabilistic statements about hook distributions in partition rows are proved.

Abstract

In this paper, we consider the asymptotic properties of hook numbers of partitions in restricted classes. More specifically, we compare the frequency with which partitions into odd parts and partitions into distinct parts have hook numbers equal to $h \geq 1$ by deriving an asymptotic formula for the total number of hooks equal to $h$ that appear among partitions into odd and distinct parts, respectively. We use these asymptotic formulas to prove a recent conjecture of the first author and collaborators that for $h \geq 2$ and $n \gg 0$, partitions into odd parts have, on average, more hooks equal to $h$ than do partitions into distinct parts. We also use our asymptotics to prove certain probabilistic statements about how hooks distribute in the rows of partitions.