{
  "id": "2006.04261",
  "version": "v1",
  "published": "2020-06-07T20:44:41.000Z",
  "updated": "2020-06-07T20:44:41.000Z",
  "title": "Combinatorics of injective words for Temperley-Lieb algebras",
  "authors": [
    "Rachael Boyd",
    "Richard Hepworth"
  ],
  "comment": "24 pages",
  "categories": [
    "math.AT",
    "math.CO",
    "math.GT"
  ],
  "abstract": "This paper studies combinatorial properties of the 'complex of planar injective words', a chain complex of modules over the Temperley-Lieb algebra that arose in our work on homological stability. Despite being a linear rather than a discrete object, our chain complex nevertheless exhibits interesting combinatorial properties. We show that the Euler characteristic of this complex is the n-th Fine number. We obtain an alternating sum formula for the representation given by its top-dimensional homology module and, under further restrictions on the ground ring, we decompose this module in terms of certain standard Young tableau. This trio of results - inspired by results of Reiner and Webb for the complex of injective words - can be viewed as an interpretation of the n-th Fine number as the 'planar' or 'Dyck path' analogue of the number derangements of n letters. This interpretation has precursors in the literature, but here emerges naturally from considerations in homological stability. Our final result shows a surprising connection between the boundary maps of our complex and the Jacobsthal numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-07T20:44:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "05E15",
      "16E40"
    ],
    "keywords": [
      "injective words",
      "temperley-lieb algebra",
      "n-th fine number",
      "combinatorics",
      "chain complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}