Generalized Pitman–Stanley Polytope: Vertices and Faces

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 \(0,1, \ldots , m\) $0,1,\dots ,m$ . 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 skew shifted shape. Moreover, we give formulas for the number of faces, as well as generating functions for the number of vertices.