{
  "id": "1811.06882",
  "version": "v2",
  "published": "2018-11-16T15:54:14.000Z",
  "updated": "2019-04-12T15:40:16.000Z",
  "title": "On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers",
  "authors": [
    "Alexander Lazar",
    "Michelle L. Wachs"
  ],
  "comment": "12 pages, 4 figures. An extended abstract, accepted to conference proceedings of Formal Power Series and Algebraic Combinatorics (FPSAC), 2019. (V2): Improvements of some results, and minor corrections",
  "categories": [
    "math.CO"
  ],
  "abstract": "Hetyei recently introduced a hyperplane arrangement (called the homogenized Linial arrangement) and used the finite field method of Athanasiadis to show that its number of regions is a median Genocchi number. These numbers count a class of permutations known as Dumont derangements. Here, we take a different approach, which makes direct use of Zaslavsky's formula relating the intersection lattice of this arrangement to the number of regions. We refine Hetyei's result by obtaining a combinatorial interpretation of the M\\\"obius function of this lattice in terms of variants of the Dumont permutations. The M\\\"obius invariant of the lattice turns out to be a (nonmedian) Genocchi number. Our techniques also yield type B, and more generally Dowling arrangement, analogs of these results.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2019-04-12T15:40:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C35",
      "05A05",
      "05A15",
      "05B35",
      "06A07",
      "11B68"
    ],
    "keywords": [
      "homogenized linial arrangement",
      "intersection lattice",
      "finite field method",
      "median genocchi number",
      "refine hetyeis result"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}