Notes on higher-dimensional 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.