Estimating the Asymptotics of Solid Partitions
Abstract
We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If denotes the number of solid partitions of an integer , we show that . This shows clear deviation from the value , attained by MacMahon numbers , that was conjectured to hold for solid partitions as well. In addition, we find estimates for other subleading terms in . In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious subleading terms. The period of the oscillatory term is proportional to , 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.
 Publication:

Journal of Statistical Physics
 Pub Date:
 February 2015
 DOI:
 10.1007/s109550141147z
 arXiv:
 arXiv:1406.5605
 Bibcode:
 2015JSP...158..950D
 Keywords:

 Solid partitions of an integer;
 Asymptotic expansion;
 Transition matrix Monte Carlo simulations;
 Condensed Matter  Statistical Mechanics;
 High Energy Physics  Theory;
 Mathematics  Combinatorics
 EPrint:
 21 pages, 8 figures