Generalized Pitman-Stanley polytope: vertices and faces

TL;DR

This paper generalizes the Pitman-Stanley polytope to include entries up to m, explores its vertices and faces, and connects it to flow polytopes, plane partitions, and Young tableaux, providing new formulas and characterizations.

Contribution

It introduces a new generalization of the Pitman-Stanley polytope, characterizes its vertices, and relates its structure to flow polytopes and combinatorial objects.

Findings

Number of vertices is a polynomial in m with leading term counting standard Young tableaux.

Provides formulas for the number of faces and generating functions for vertices.

Shows the generalized polytope can be realized as a flow polytope of a grid graph.

Abstract

In 1999, Pitman and Stanley introduced the polytope bearing their name along with a study of its faces, lattice points, and volume. The Pitman-Stanley polytope is well-studied due to its connections to probability, parking functions, the generalized permutahedra, and flow polytopes. Its lattice points correspond to plane partitions of skew shape with entries 0 and 1. Pitman and Stanley remarked that their polytope can be generalized so that lattice points correspond to plane partitions of skew shape with entries $0,1, \ldots , m$. Since then, this generalization has been untouched. We study this generalization and show that it can also be realized as a flow polytope of a grid graph. We give multiple characterizations of its vertices in terms of plane partitions of skew shape and integer flows. For a fixed skew shape, we show that the number of vertices of this polytope is a polynomial in $m$ whose leading term, in certain cases, counts standard Young tableaux of a shifted shape. Moreover, we give formulas for the number of faces, as well as generating functions for the number of vertices.