{
  "id": "1503.06243",
  "version": "v1",
  "published": "2015-03-20T22:34:26.000Z",
  "updated": "2015-03-20T22:34:26.000Z",
  "title": "Face rings of cycles, associahedra, and standard Young tableaux",
  "authors": [
    "Anton Dochtermann"
  ],
  "comment": "14 pages, 4 figures",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "We show that J_n, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n-3)-dimensional simplicial associahedron A_n. This resolution is not minimal for n > 5; in this case the Betti numbers of J_n are strictly smaller than the f-vector of A_n. We show that in fact the Betti numbers of J_n are in bijection with the number of standard Young tableaux of shape (d+1, 2, 1^{n-d-3}). This complements the fact that the number of (d-1)-dimensional faces of A_n are given by the number of standard Young tableaux of (super)shape (d+1, d+1, 1^{n-d-3}); a bijective proof of this result was first provided by Stanley. An application of discrete Morse theory yields a cellular resolution of J_n that we show is minimal at the first syzygy. We furthermore exhibit a simple involution on the set of associahedron tableaux with fixed points given by the Betti tableaux, suggesting a Morse matching and in particular a poset structure on these objects.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-20T22:34:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E40",
      "13D02",
      "05E45",
      "13F55",
      "52B05"
    ],
    "keywords": [
      "standard young tableaux",
      "face rings",
      "associahedron",
      "betti numbers",
      "discrete morse theory yields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150306243D"
    }
  }
}