Notes on higher-dimensional partitions

TL;DR

This paper introduces a series of transforms that reveal underlying structures of higher-dimensional partitions, leading to a significant reduction in computational complexity and enabling enumeration of partitions up to size 25 in any dimension.

Contribution

It presents new transforms and a triangle F that simplify the calculation of higher-dimensional partitions, connecting combinatorial structures with efficient enumeration methods.

Findings

Existence of transforms capturing structures of higher-dimensional partitions.

Development of a triangle F reducing data needed for partition counts by half.

Successful enumeration of partitions up to size 25 in any dimension.

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.