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 pmod6. For a positive integer r, let Ar(n) count the partitions of n in which every part occurring an odd number of times occurs at least 2r+1 times, and let Br(n) count the partitions into parts that are even or congruent to 2r+1 pmod4r+2. Andrews proved that Ar(n)=Br(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 Br-type partitions exhibit more $1$-hooks than the Ar-type partitions, whereas for hook lengths $2$ and $3$, the bias is reversed when r=1 and favors the Br-type partitions when r≥ 2. We also establish asymptotic inequalities among hook numbers of different lengths within each partition class. Finally, for fixed hook lengths ℓ∈{1,2,3}, we prove that the Br-type partitions exhibit more ℓ-hooks than the Br+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.
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BibTeX
@misc{barman2026inequalities,
title = {{Inequalities and asymptotics for hook numbers in two partition functions of MacMahon and Andrews}},
author = {Rupam Barman and Archit Deb and Sayantan Ghorai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33791},
eprint = {2610.33791},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33791},
}
RIS
TY - UNPB
TI - Inequalities and asymptotics for hook numbers in two partition functions of MacMahon and Andrews
AU - Rupam Barman
AU - Archit Deb
AU - Sayantan Ghorai
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33791
UR - https://arxiv.org/abs/2610.33791
AB - 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 pmod6. 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 pmod4r+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≥ 2. We also establish asymptotic inequalities among hook numbers of different lengths within each partition class. Finally, for fixed hook lengths ℓ∈{1,2,3}, we prove that the B_r-type partitions exhibit more ℓ-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.
ER -