{
  "id": "2006.07915",
  "version": "v1",
  "published": "2020-06-14T14:33:47.000Z",
  "updated": "2020-06-14T14:33:47.000Z",
  "title": "Inversion arrangements and the weak Bruhat order",
  "authors": [
    "Neil J. Y. Fan"
  ],
  "comment": "6 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "For each permutation $w$, we can construct a collection of hyperplanes $\\mathcal{A}_w$ according to the inversions of $w$, which is called the inversion hyperplane arrangement associated to $w$. It was conjectured by Postnikov and confirmed by Hultman, Linusson, Shareshian and Sj\\\"{o}strand that the number of regions of $\\mathcal{A}_w$ is less than or equal to the number of permutations below $w$ in the Bruhat order, with the equality holds if and only if $w$ avoids the four patterns 4231, 35142, 42513 and 351624. In this paper, we show that the number of regions of $\\mathcal{A}_w$ is greater than or equal to the number of permutations below $w$ in the weak Bruhat order, with the equality holds if and only if $w$ avoids the patterns 231 and 312.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-14T14:33:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak bruhat order",
      "inversion arrangements",
      "equality holds",
      "inversion hyperplane arrangement",
      "permutation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}