Notes on higher-dimensional partitions
arXiv:1203.4419 · doi:10.1016/j.jcta.2012.11.005
Abstract
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 triangles 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 triangle, 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 triangle F. Using modifications of an algorithm due to Bratley-McKay, we are able to directly enumerate entries in some of the triangles. As a result, we have been able to compute numbers of partitions of positive integers <= 25 in any dimension.
36 pages; Mathematica file attached; See http://www.physics.iitm.ac.in/~suresh/partitions.html to generate numbers of partitions