MacMahon's ${\mathrm{\Omega }}_{\ge }$ 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 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.