{
  "id": "2010.00930",
  "version": "v1",
  "published": "2020-10-02T11:47:41.000Z",
  "updated": "2020-10-02T11:47:41.000Z",
  "title": "The Bernardi formula for non-transitive deformations of the braid arrangement",
  "authors": [
    "Ankit Bisain",
    "Eric J. Hanson"
  ],
  "comment": "16 pages, 8 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Bernardi has given a general formula to compute the number of regions of a deformation of the braid arrangement as a signed sum over boxed trees. We prove that the contribution to this sum of the set of boxed trees sharing an underlying rooted labeled tree is -1, 0, or 1 and give an algorithm for computing this value. We then restrict to arrangements which we call almost transitive and construct a sign-reversing involution which reduces Bernardi's signed sum to the enumeration of a set of rooted labeled trees in this case. We conclude by explicitly enumerating the trees corresponding to the regions of certain nested Ish arrangements which we call non-negative, recovering their known counting formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-02T11:47:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19"
    ],
    "keywords": [
      "braid arrangement",
      "bernardi formula",
      "non-transitive deformations",
      "rooted labeled tree",
      "boxed trees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}