Generalized Pitman-Stanley polytope: vertices and faces
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arXiv
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| Format: | Preprint |
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2023
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| author | Dugan, William T. Hegarty, Maura Morales, Alejandro H. Raymond, Annie |
| author_facet | Dugan, William T. Hegarty, Maura Morales, Alejandro H. Raymond, Annie |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_09925 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized Pitman-Stanley polytope: vertices and faces Dugan, William T. Hegarty, Maura Morales, Alejandro H. Raymond, Annie Combinatorics Primary: 05C21, 52B05, 05A15, Secondary: 05A19, 06A07, 52B20 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. |
| title | Generalized Pitman-Stanley polytope: vertices and faces |
| topic | Combinatorics Primary: 05C21, 52B05, 05A15, Secondary: 05A19, 06A07, 52B20 |
| url | https://arxiv.org/abs/2307.09925 |