Mahonian Partition Identities via Polyhedral Geometry

In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon’s Ω operator to systematically compute generating functions \({\sum \nolimits }_{\lambda \in P}{z}_{1}^{{\lambda }_{1}}\ldots {z}_{n}^{{\lambda }_{n}}\) for some set P of integer partitions λ = (λ1, …, λ n ). Our goal is to geometrically prove and extend many of Andrews et al.’s theorems, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron.