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

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

2003-03-28

arxiv.org/abs/cond-mat/0303607v1

Journal of Physics A, Vol 36, 6651 (2003)

Physics

Condensed Matter

Statistical Mechanics

6 pages, 4 figures, revtex4

Scientific paper

10.1088/0305-4470/36/24/304

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.

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