{
  "id": "2309.14261",
  "version": "v1",
  "published": "2023-09-25T16:19:48.000Z",
  "updated": "2023-09-25T16:19:48.000Z",
  "title": "The $s$-weak order and $s$-permutahedra II: The combinatorial complex of pure intervals",
  "authors": [
    "Cesar Ceballos",
    "Viviane Pons"
  ],
  "comment": "45 pages, 28 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "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 $\\nu$-associahedron. Finally, we present three polytopality conjectures, evidence supporting them, and some hints about potential generalizations to other finite Coxeter groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-25T16:19:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "06B05",
      "06B10",
      "52B05"
    ],
    "keywords": [
      "weak order",
      "combinatorial complex",
      "pure intervals",
      "permutahedron",
      "tamari lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}