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

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.

arXiv comment6 pages, 4 figures, revtex4

References in corpus (3)

• An efficient, multiple range random walk algorithm to calculate the density of states
• Flat Histogram Method of Wang-Landau and N-fold Way
• Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions

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