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arXiv · 2307.09925

Generalized Pitman-Stanley polytope: vertices and faces

William T. Dugan·Maura Hegarty·Alejandro H. Morales·Annie Raymond
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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.

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William T. Dugan, Maura Hegarty, Alejandro H. Morales, Annie Raymond. 2023-07-19. Generalized Pitman-Stanley polytope: vertices and faces. https://doi.org/10.1007/s00454-024-00704-3

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