Mahonian Partition Identities Via Polyhedral Geometry

TL;DR

This paper uses polyhedral geometry to provide geometric proofs and extensions of Mahonian partition identities originally studied through MacMahon's Omega operator, connecting partition generating functions with lattice points in polyhedra.

Contribution

It introduces a geometric approach to prove and extend Mahonian partition identities by representing partition sets as lattice points in polyhedra.

Findings

Geometric proofs of Mahonian identities

Extension of partition identities via polyhedral methods

Connection between generating functions and lattice points

Abstract

In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's $\Omega$ operator to systematically compute generating functions $\sum_{\la \in P} z_1^{\la_1}...z_n^{\la_n}$ for some set $P$ of integer partitions $\la = (\la_1,..., \la_n)$. Our goal is to geometrically prove and extend many of the Andrews et al theorems, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron.