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Enumeration of partitions via socle reduction

Michele Graffeo, Sergej Monavari, Riccardo Moschetti, Andrea T. Ricolfi

TL;DR

This work develops a refined framework for enumerating higher dimensional partitions by introducing socle reduction, which partitions the counting problem into simpler blocks classified by embedding dimension and socle degree. It proves that refined generating functions, notably $\,\mathsf Y_e(t)$ and $\mathsf C_x(t)$, are rational and connects them via inversion formulas to reconstruct the full counts $p_d^n$, enabling exact computations up to size $d\le 30$. The authors define and count M-partitions, hydral partitions, and headstrong M-partitions, deriving exact generating functions in several families and providing an algorithmic pipeline (implemented in Sage, Mathematica, and Macaulay2) to compute the numbers efficiently. They also uncover deep connections to geometry, particularly Quot schemes, and pose conjectures regarding MacMahon’s discrepancy and sparsity patterns, with implications for the structure of partition counts in higher dimensions.

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.

Enumeration of partitions via socle reduction

TL;DR

This work develops a refined framework for enumerating higher dimensional partitions by introducing socle reduction, which partitions the counting problem into simpler blocks classified by embedding dimension and socle degree. It proves that refined generating functions, notably and , are rational and connects them via inversion formulas to reconstruct the full counts , enabling exact computations up to size . The authors define and count M-partitions, hydral partitions, and headstrong M-partitions, deriving exact generating functions in several families and providing an algorithmic pipeline (implemented in Sage, Mathematica, and Macaulay2) to compute the numbers efficiently. They also uncover deep connections to geometry, particularly Quot schemes, and pose conjectures regarding MacMahon’s discrepancy and sparsity patterns, with implications for the structure of partition counts in higher dimensions.

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.
Paper Structure (29 sections, 28 theorems, 172 equations, 7 figures, 2 tables)

This paper contains 29 sections, 28 theorems, 172 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Refined partition counting: socle reduction technique
  4. Exact embedding dimension
  5. No socle in degree 1
  6. No socle in degree 2
  7. Counting M-partitions
  8. Geometry and Physics
  9. Background and notation
  10. Conventions
  11. Higher dimensional partitions
  12. Apolarity
  13. Monomial ideals
  14. Borel transform
  15. Generating functions of partitions
  16. ...and 14 more sections

Key Result

Theorem A

Fix $e \in {\mathbb{Z}}_{>0}$. Then in ${\mathbb{Z}}(t)$, where is a polynomial of degree at most $2e-1$, whose coefficients are given by for $0\leqslant h \leqslant 2e-1$. In particular, $\mathsf Y_e(t)$ is a rational function.

Figures (7)

  • Figure 1: A partition $\lambda\in \mathrm{P}^3_{16}$ counted by $c_{12,(3,4,8)}^3$.
  • Figure 2: A partition $\lambda'\in \mathrm{P}^3_{13}$ counted by $c_{9,(2,3,5)}^3$.
  • Figure 3: Examples of partitions from \ref{['thm:expexp']} in the case $s=3$ and $n=2,3$.
  • Figure 4: On the left $(1,1) \in \lambda$, on the right $(1,1)\notin \lambda$.
  • Figure 5: Examples of planar and (a projection of) a solid headstrongs partitions.
  • ...and 2 more figures

Theorems & Definitions (75)

  • Theorem A: \ref{['thm:geny']}
  • Theorem B: \ref{['thm:genc']}
  • Proposition 1.1: \ref{['prop:recur']}
  • Theorem C: \ref{['thm: hydral structure', 'cor: count hydral']}
  • Theorem D: \ref{['thm:expexp']}
  • Definition 2.1
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 65 more