Generalized Pitman–Stanley Polytope: Vertices and Faces
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Author(s)
Dugan, William T.
Hegarty, Maura
Morales, Alejandro H.
Raymond, Annie
Date Issued
December 9, 2024
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Dugan, W.T., Hegarty, M., Morales, A.H. et al. Generalized Pitman–Stanley Polytope: Vertices and Faces. Discrete Comput Geom 74, 492–543 (2025).
Version
Author's final manuscript
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 , … , 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.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00454-024-00704-3
