{
  "id": "1010.5614",
  "version": "v1",
  "published": "2010-10-27T09:10:59.000Z",
  "updated": "2010-10-27T09:10:59.000Z",
  "title": "Enumeration of linear chord diagrams",
  "authors": [
    "J. E. Andersen",
    "R. C. Penner",
    "C. M. Reidys",
    "M. S. Waterman"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A linear chord diagram canonically determines a fatgraph and hence has an associated genus $g$. We compute the natural generating function ${\\bf C}_g(z)=\\sum_{n\\geq 0} {\\bf c}_g(n)z^n$ for the number ${\\bf c}_g(n)$ of linear chord diagrams of fixed genus $g\\geq 1$ with a given number $n\\geq 0$ of chords and find the remarkably simple formula ${\\bf C}_g(z)=z^{2g}R_g(z) (1-4z)^{{1\\over 2}-3g}$, where $R_g(z)$ is a polynomial of degree at most $g-1$ with integral coefficients satisfying $R_g({1\\over 4})\\neq 0$ and $R_g(0) = {\\bf c}_g(2g)\\neq 0.$ In particular, ${\\bf C}_g(z)$ is algebraic over $\\mathbb C(z)$, which generalizes the corresponding classical fact for the generating function ${\\bf C}_0(z)$ of the Catalan numbers. As a corollary, we also calculate a related generating function germaine to the enumeration of knotted RNA secondary structures, which is again found to be algebraic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-27T09:10:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "enumeration",
      "linear chord diagram canonically determines",
      "knotted rna secondary structures",
      "remarkably simple formula",
      "natural generating function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.5614A"
    }
  }
}