Inequalities and asymptotics for hook numbers in two partition functions of MacMahon and Andrews

Abstract

Motivated by recent studies of hook length biases in restricted partitions, we investigate the total number of hooks of fixed length in two partition functions introduced by MacMahon and Andrews. In $1916$, MacMahon proved that the number of partitions of $n$ in which no part appears with multiplicity one equals the number of partitions of $n$ into parts that are even or congruent to $3 \pmod{6}$. For a positive integer $r$, let $A_r(n)$ count the partitions of $n$ in which every part occurring an odd number of times occurs at least $2r+1$ times, and let $B_r(n)$ count the partitions into parts that are even or congruent to $2r+1 \pmod{4r+2}$. Andrews proved that $A_r(n)=B_r(n)$ for all positive integers $r$ and nonnegative integers $n$, thereby recovering MacMahon's identity as the special case $r=1$. We derive generating functions and asymptotic formulas for the total numbers of hooks of lengths $1$, $2$, and $3$ in both partition classes, employing Wright's circle method. These asymptotic formulas yield several hook length bias results. In particular, for sufficiently large $n$, the $B_r$-type partitions exhibit more $1$-hooks than the $A_r$-type partitions, whereas for hook lengths $2$ and $3$, the bias is reversed when $r=1$ and favors the $B_r$-type partitions when $r\geq 2$. We also establish asymptotic inequalities among hook numbers of different lengths within each partition class. Finally, for fixed hook lengths $\ell\in\{1,2,3\}$, we prove that the $B_r$-type partitions exhibit more $\ell$-hooks than the $B_{r+1}$-type partitions for sufficiently large $n$. These results reveal systematic asymptotic patterns in the distribution of small hook lengths across the two partition families.