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

TL;DR

This paper develops a computational framework for MacMahon's partition analysis, enabling simplified solutions to classical combinatorial problems and addressing complex new challenges.

Contribution

It introduces a fundamental computational method for MacMahon's partition analysis, improving problem-solving efficiency in combinatorics.

Findings

Simplified computations for Han's formula

Efficient solutions for k-gon partitions

Successful application to a complex problem

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.