Enumeration of partitions via socle reduction
Combinatorics
2025-01-20 v1 Commutative Algebra
Algebraic Geometry
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.
Cite
@article{arxiv.2501.10267,
title = {Enumeration of partitions via socle reduction},
author = {Michele Graffeo and Sergej Monavari and Riccardo Moschetti and Andrea T. Ricolfi},
journal= {arXiv preprint arXiv:2501.10267},
year = {2025}
}
Comments
48 pages. Ancillary files. Comments welcome