Estimating the asymptotics of integer partitions in intermediate dimensions (d = 3,4,5,6)
Abstract
It was recently shown by Yeliussizov \cite{Yeliussizov} that integer partitions in dimensions d ≥ 7 asymptotically grow strictly faster than MacMahon numbers. As MacMahon numbers match with integer partitions in dimensions d = 1,2, the comparison of asymptotics of integer partitions with MacMahon numbers in intermediate dimensions (d = 3,4,5,6) is an open question. In this work, we perform Markov chain Monte Carlo (MCMC) simulations till N=15000 by using adaptive weight learning followed by conventional MCMC steps to numerically estimate the asymptotics of integer partitions in these intermediate dimensions. We numerically establish that in these intermediate dimensions, partitions asymptotically grow faster than MacMahon numbers. More specifically, assuming that the limits exist, we show: limn→∞n−3/4log p3(n) = 1.8196 ± 0.0019, limn→∞n−4/5log p4(n) = 1.7215 ± 0.0045, limn→∞n−5/6log p5(n) = 1.6521 ± 0.0059, and limn→∞log n−6/7p6(n) = 1.652 ± 0.021 for partitions in dimensions d=3,4,5, and $6$ respectively. These numbers are all larger than MacMahon leading order asymptotic coefficients of $1.7898, 1.6614, 1.5737,$ and $1.509$ respectively. Additionally, we also find estimates for some of the sub-leading asymptotic terms in log pd(n) in each of the dimensions.
Related
- Excedances and Deficiencies of Integer Partitions
- Order, Structure and Reality: Hierarchical Shell Structure of Integer Partitions
- Asymptotics for partitions over the Fibonacci numbers and related sequences, II
- Intersecting integer partitions: star bounds and counterexamples at every scale
- On the Krull dimensions of rings of integer-valued polynomials and polynomial rings
BibTeX
@misc{mondal2026estimating,
title = {{Estimating the asymptotics of integer partitions in intermediate dimensions ($d = 3,4,5,6$)}},
author = {Avinandan Mondal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03034},
eprint = {2609.03034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03034},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Estimating the asymptotics of integer partitions in intermediate dimensions (d = 3,4,5,6)
AU - Avinandan Mondal
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.03034
UR - https://arxiv.org/abs/2609.03034
AB - It was recently shown by Yeliussizov \cite{Yeliussizov} that integer partitions in dimensions d ≥ 7 asymptotically grow strictly faster than MacMahon numbers. As MacMahon numbers match with integer partitions in dimensions d = 1,2, the comparison of asymptotics of integer partitions with MacMahon numbers in intermediate dimensions (d = 3,4,5,6) is an open question. In this work, we perform Markov chain Monte Carlo (MCMC) simulations till N=15000 by using adaptive weight learning followed by conventional MCMC steps to numerically estimate the asymptotics of integer partitions in these intermediate dimensions. We numerically establish that in these intermediate dimensions, partitions asymptotically grow faster than MacMahon numbers. More specifically, assuming that the limits exist, we show: lim_n→∞n^−3/4log p_3(n) = 1.8196 ± 0.0019, lim_n→∞n^−4/5log p_4(n) = 1.7215 ± 0.0045, lim_n→∞n^−5/6log p_5(n) = 1.6521 ± 0.0059, and lim_n→∞log n^−6/7p_6(n) = 1.652 ± 0.021 for partitions in dimensions d=3,4,5, and $6$ respectively. These numbers are all larger than MacMahon leading order asymptotic coefficients of $1.7898, 1.6614, 1.5737,$ and $1.509$ respectively. Additionally, we also find estimates for some of the sub-leading asymptotic terms in log p_d(n) in each of the dimensions.
ER -