On the asymptotics of higher-dimensional partitions

TL;DR

This paper investigates the asymptotic behavior of higher-dimensional partitions, conjecturing they mirror the known asymptotics of three-dimensional MacMahon numbers, supported by enumeration data and preliminary evidence for dimensions four and five.

Contribution

It proposes a conjecture that higher-dimensional partitions share asymptotic behavior with three-dimensional MacMahon numbers, supported by enumeration and preliminary evidence.

Findings

Enumeration of solid partitions up to 68 shows accurate asymptotic predictions.

Asymptotic formulas with adjustable parameters improve fit with data.

Initial evidence suggests similar behavior extends to four and five dimensions.

Abstract

We conjecture that the asymptotic behavior of the numbers of solid (three-dimensional) partitions is identical to the asymptotics of the three-dimensional MacMahon numbers. Evidence is provided by an exact enumeration of solid partitions of all integers <=68 whose numbers are reproduced with surprising accuracy using the asymptotic formula (with one free parameter) and better accuracy on increasing the number of free parameters. We also conjecture that similar behavior holds for higher-dimensional partitions and provide some preliminary evidence for four and five-dimensional partitions.