We conjecture that the asymptotic behavior of the numbers of solid (three-dimensional) partitions is identical to the asymptotics of the three-dimensional MacMahon numbers. Evidence is provided by an exact enumeration of solid partitions of all integers ≤ 68 whose numbers are reproduced with surprising accuracy using the asymptotic formula (with one free parameter) and better accuracy on increasing the number of free parameters. We also conjecture that similar behavior holds for higher dimensional partitions and provides some preliminary evidence for four- and five-dimensional partitions. © 2012 IOP Publishing Ltd.

Suresh Govindarajan

Department Of Physics

Srivatsan Balakrishnan

Naveen S Prabhakar

Fulltext

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Journal of Physics A: Mathematical and Theoretical

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. © 2014, Springer Science+Business Media New York.

Journal of Statistical Physics

Estimating the Asymptotics of Solid Partitions

We show the existence of a series of transforms that capture several structures that underlie higher-dimensional partitions. These transforms lead to a sequence of matrices whose entries are given combinatorial interpretations as the number of particular types of skew Ferrers diagrams. The end result of our analysis is the existence of a matrix, that we denote by F, which implies that the data needed to compute the number of partitions of a given positive integer is reduced by a factor of half. The number of spanning rooted forests appears intriguingly in a family of entries in the matrix, F. Using modifications of an algorithm due to Bratley-McKay, we are able to directly enumerate entries in some of the matrices. As a result, we have been able to compute numbers of partitions of positive integers ≤26 in any dimension. © 2012 Elsevier Inc.

Journal	Journal of Physics A: Mathematical and Theoretical
Publisher	IOP Science
ISSN	17518113
Open Access	No