Publication: Estimating the Asymptotics of Solid Partitions
Date
01-02-2015
Authors
Suresh Govindarajan
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Abstract
We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If p3(n) denotes the number of solid partitions of an integer n, we show that (Formula presented.). This shows clear deviation from the value 1.7898, attained by MacMahon numbers m3(n), that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in log p3(n). In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious sub-leading terms. The period of the oscillatory term is proportional to n1/4, the natural scale in the problem. This new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function.
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Keywords
Asymptotic expansion, Solid partitions of an integer, Transition matrix Monte Carlo simulations