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MacMahon's $Ω_\geq$ operator: A computational framework

Feihu Liu, Guoce Xin

TL;DR

The paper develops a fundamental computational method for MacMahon's partition analysis by combining iterated Laurent series with partial fraction decomposition to evaluate the Omega_ge operator on Elliott rational functions. This approach yields a practical framework for eliminating lambda variables and producing closed-form generating functions, demonstrated through Han's formula, the k-gon partition problem, the two-dimensional problem, and a notably hard generating function case, along with explicit generating-function relations. The work unifies and simplifies multiple classic results while providing a Maple/algorithmic blueprint that can aid lattice-point enumeration and Ehrhart-type analyses. Overall, it offers a concrete, implementable methodology that bridges theory and computational practice in partition analysis.

Abstract

MacMahon introduced partition analysis in his book ``Combinatory Analysis'' as a computational technique for solving problems related to systems of linear Diophantine equations and inequalities. This paper aims to develop a fundamental computational method for MacMahon's partition analysis. As applications, we present simplified computations for ``Han's formula'', the ``$k$-gon partitions problem'', and the ``two-dimensional problem''. Moreover, we apply our method to solve a challenging problem.

MacMahon's $Ω_\geq$ operator: A computational framework

TL;DR

The paper develops a fundamental computational method for MacMahon's partition analysis by combining iterated Laurent series with partial fraction decomposition to evaluate the Omega_ge operator on Elliott rational functions. This approach yields a practical framework for eliminating lambda variables and producing closed-form generating functions, demonstrated through Han's formula, the k-gon partition problem, the two-dimensional problem, and a notably hard generating function case, along with explicit generating-function relations. The work unifies and simplifies multiple classic results while providing a Maple/algorithmic blueprint that can aid lattice-point enumeration and Ehrhart-type analyses. Overall, it offers a concrete, implementable methodology that bridges theory and computational practice in partition analysis.

Abstract

MacMahon introduced partition analysis in his book ``Combinatory Analysis'' as a computational technique for solving problems related to systems of linear Diophantine equations and inequalities. This paper aims to develop a fundamental computational method for MacMahon's partition analysis. As applications, we present simplified computations for ``Han's formula'', the ``-gon partitions problem'', and the ``two-dimensional problem''. Moreover, we apply our method to solve a challenging problem.

Paper Structure

This paper contains 10 sections, 14 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Fundamental Calculation Method
  3. Fundamental Formulas
  4. A Computational Framework
  5. Applications
  6. Han's Formula
  7. $k$-gon Partitions
  8. Two Dimensional Problem
  9. A Hard Problem
  10. Two Generating Functions

Key Result

Theorem 2.1

Assume that the partial fraction decomposition of $E(\lambda)$ is given by where each $u_i$ is independent of $\lambda$; $P(\lambda)$, $p(\lambda)$, and $A_i(\lambda)$ are polynomials, $\deg p(\lambda) < s$, and $\deg A_i(\lambda) < a_i$ for all $i$. Then,

Theorems & Definitions (30)

  • Definition 1.1
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Example 2.5
  • Proposition 3.1
  • ...and 20 more