Enumeration of partitions via socle reduction

TL;DR

This paper investigates the enumeration of higher dimensional partitions, establishing equivalences with simpler classes, deriving formulas for their generating functions, and providing a procedure for their enumeration.

Contribution

It introduces a new approach to count higher dimensional partitions by reducing the problem to simpler classes with specific constraints, and provides explicit formulas and enumeration procedures.

Findings

Exact formulas for generating functions of certain partition classes

A procedure for enumerating partitions in general cases

Number of partitions up to size 30 in any dimension determined

Abstract

We study the enumeration problem of higher dimensional partitions, a natural generalisation of classical integer partitions. We show that their counting problem is equivalent to the enumeration of simpler classes of higher dimensional partitions, satisfying suitable constraints on their embedding dimension and socle type. We provide exact formulas for the generating functions of several infinite families of such partitions, and design a procedure enumerating them in the general case. As a proof of concept, we determine the number of partitions of size up to 30 in any dimension.