Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer

TL;DR

This paper investigates the asymptotic behavior of solid partitions of integers using numerical methods, providing evidence that supports a conjectured leading order growth rate for the number of such partitions.

Contribution

It introduces numerical estimation techniques for solid partitions up to 8000 and offers new insights into their asymptotic growth, supporting the MacMahon conjecture.

Findings

ln[p(n)]/n^(3/4) ≈ 1.79 for large n

Numerical methods extend to n=8000

Supports the MacMahon conjecture's asymptotic form

Abstract

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where p(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.