The $s$-weak order and $s$-permutahedra II: The combinatorial complex of pure intervals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929730414968832 |
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| author | Ceballos, Cesar Pons, Viviane |
| author_facet | Ceballos, Cesar Pons, Viviane |
| contents | This paper introduces the geometric foundations for the study of the $s$-permutahedron and the $s$-associahedron, two objects that encode the underlying geometric structure of the $s$-weak order and the $s$-Tamari lattice. We introduce the $s$-permutahedron as the complex of pure intervals of the $s$-weak order, present enumerative results about its number of faces, and prove that it is a combinatorial complex. This leads, in particular, to an explicit combinatorial description of the intersection of two faces. We also introduce the $s$-associahedron as the complex of pure $s$-Tamari intervals of the $s$-Tamari lattice, show some enumerative results, and prove that it is isomorphic to a well chosen $ν$-associahedron. Finally, we present three polytopality conjectures, evidence supporting them, and some hints about potential generalizations to other finite Coxeter groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_14261 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The $s$-weak order and $s$-permutahedra II: The combinatorial complex of pure intervals Ceballos, Cesar Pons, Viviane Combinatorics Primary 20F55, 06B05 and 06B10, Secondary 52B05 This paper introduces the geometric foundations for the study of the $s$-permutahedron and the $s$-associahedron, two objects that encode the underlying geometric structure of the $s$-weak order and the $s$-Tamari lattice. We introduce the $s$-permutahedron as the complex of pure intervals of the $s$-weak order, present enumerative results about its number of faces, and prove that it is a combinatorial complex. This leads, in particular, to an explicit combinatorial description of the intersection of two faces. We also introduce the $s$-associahedron as the complex of pure $s$-Tamari intervals of the $s$-Tamari lattice, show some enumerative results, and prove that it is isomorphic to a well chosen $ν$-associahedron. Finally, we present three polytopality conjectures, evidence supporting them, and some hints about potential generalizations to other finite Coxeter groups. |
| title | The $s$-weak order and $s$-permutahedra II: The combinatorial complex of pure intervals |
| topic | Combinatorics Primary 20F55, 06B05 and 06B10, Secondary 52B05 |
| url | https://arxiv.org/abs/2309.14261 |