Numerical estimation of the asymptotic behaviour of solid partitions of an integer
V Mustonen, R Rajesh - Journal of Physics A: Mathematical and …, 2003 - iopscience.iop.org
Journal of Physics A: Mathematical and General, 2003•iopscience.iop.org
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 lim n→∞ ln (p 3 (n))/n 3/4=
1.79±0.01, where p 3 (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 …
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 lim n→∞ ln (p 3 (n))/n 3/4=
1.79±0.01, where p 3 (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 …
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 lim n→∞ ln (p 3 (n))/n 3/4= 1.79±0.01, where p 3 (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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