{
  "id": "1707.04351",
  "version": "v1",
  "published": "2017-07-13T23:02:01.000Z",
  "updated": "2017-07-13T23:02:01.000Z",
  "title": "A Generating Function for the Distribution of Runs in Binary Words",
  "authors": [
    "James J. Madden"
  ],
  "comment": "5 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $N(n,r,k)$ denote the number of binary words of length $n$ that begin with $0$ and contain exactly $k$ runs (i.e., maximal subwords of identical consecutive symbols) of length $r$. We show that the generating function for the sequence $N(n,r,0)$, $n=0,1,\\ldots$, is $(1-x)(1-2x + x^r-x^{r+1})^{-1}$ and that the generating function for $\\{N(n,r,k)\\}$ is $x^{kr}$ time the $k+1$ power of this. We extend to counts of words containing exactly $k$ runs of $1$s by using symmetries on the set of binary words.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-13T23:02:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "60C05"
    ],
    "keywords": [
      "binary words",
      "generating function",
      "distribution",
      "maximal subwords",
      "identical consecutive symbols"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}