MacMahon's Ω_≥ Operator: A Computational Framework

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 framework 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.