Partial permutation and alternating sign matrix polytopes

TL;DR

This paper introduces and analyzes a new class of polytopes formed from partial alternating sign matrices, exploring their geometric properties, face structures, and connections to partial permutohedra, including volume conjectures.

Contribution

It defines partial permutation and alternating sign matrix polytopes, characterizes their facets and face lattices, and links them to partial permutohedra with new enumeration and volume results.

Findings

Determined inequality descriptions and face lattices of the polytopes

Connected partial permutohedra to projections of these polytopes

Provided volume conjecture for fixed parameters

Abstract

We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial permutohedra that we show arise naturally as projections of these polytopes. We enumerate facets and also characterize the face lattices of partial permutohedra in terms of chains in the Boolean lattice. Finally, we have a result and a conjecture on the volume of partial permutohedra when one parameter is fixed to be two.