{
  "id": "1405.1608",
  "version": "v3",
  "published": "2014-05-07T14:07:47.000Z",
  "updated": "2015-08-28T16:33:54.000Z",
  "title": "Combinatorics of diagrams of permutations",
  "authors": [
    "Joel Brewster Lewis",
    "Alejandro H. Morales"
  ],
  "comment": "v3: 29 pages, 11 figures. Addressed minor suggestions from referees, typos fixed, and updated references. v2: 29 pages, 11 figures. Part of the proof of Theorem 2.1 has been replaced by an elegant argument by Axel Hultman, which is now included as Appendix A. v1: 31 pages, 11 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "There are numerous combinatorial objects associated to a Grassmannian permutation $w_\\lambda$ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their $q$-analogues in the case of permutations $w$ that are not necessarily Grassmannian. We give two main results: first, we show that certain acyclic orientations, rook placements avoiding a diagram of $w$, and fillings of a diagram of $w$ are equinumerous for all permutations $w$. Second, we give a $q$-analogue of a result of Hultman-Linusson-Shareshian-Sj\\\"ostrand by showing that under a certain pattern condition the Poincar\\'e polynomial for the Bruhat interval of $w$ essentially counts invertible matrices avoiding a diagram of $w$ over a finite field. In addition to our main results, we include at the end a number of open questions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-07-10T18:28:39.000Z",
      "comment": "v2: 29 pages, 11 figures. Part of the proof of Theorem 2.1 has been replaced by an elegant argument by Axel Hultman, which is now included as Appendix A. v1: 31 pages, 11 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-08-28T16:33:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15",
      "05A19",
      "05A30"
    ],
    "keywords": [
      "counts invertible matrices avoiding",
      "combinatorics",
      "main results",
      "acyclic orientations",
      "numerous combinatorial objects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.1608B"
    }
  }
}