{
  "id": "2212.11556",
  "version": "v1",
  "published": "2022-12-22T09:22:38.000Z",
  "updated": "2022-12-22T09:22:38.000Z",
  "title": "$s$-week order and $s$-permutahedra I: combinatorics and lattice structure",
  "authors": [
    "Cesar Ceballos",
    "Viviane Pons"
  ],
  "comment": "35 pages, 17 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "This is the first contribution of a sequence of papers introducing the notions of $s$-weak order and $s$-permutahedra, certain discrete objects that are indexed by a sequence of non-negative integers $s$. In this first paper, we concentrate purely on the combinatorics and lattice structure of the $s$-weak order, a partial order on certain decreasing trees which generalizes the classical weak order on permutations. In particular, we show that the $s$-weak order is a semidistributive and congruence uniform lattice, generalizing known results for the classical weak order on permutations. Restricting the $s$-weak order to certain trees gives rise to the $s$-Tamari lattice, a sublattice which generalizes the classical Tamari lattice. We show that the $s$-Tamari lattice can be obtained as a quotient lattice of the $s$-weak order when $s$ has no zeros, and show that the $s$-Tamari lattices (for arbitrary $s$) are isomorphic to the $\\nu$-Tamari lattices of Pr\\'eville-Ratelle and Viennot. The underlying geometric structure of the $s$-weak order will be studied in a sequel of this paper, where we introduce the notion of $s$-permutahedra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-22T09:22:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "06B05",
      "06B10",
      "52B05",
      "G.2.1"
    ],
    "keywords": [
      "lattice structure",
      "week order",
      "permutahedra",
      "combinatorics",
      "classical weak order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}