{
  "id": "2306.07870",
  "version": "v1",
  "published": "2023-06-13T15:59:39.000Z",
  "updated": "2023-06-13T15:59:39.000Z",
  "title": "Subsequence frequency in binary words",
  "authors": [
    "Krishna Menon",
    "Anurag Singh"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The numbers we study in this paper are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that contain the word $p$ (as a subsequence) exactly $k$ times. Our motivation comes from the analogous study of pattern containment in permutations. In our first set of results, we obtain explicit expressions for $B_{n, p}(k)$ for small values of $k$. We then focus on words $p$ with at most $3$ runs and study the maximum number of occurrences of $p$ a word of length $n$ can have. We also study the internal zeros in the sequence $(B_{n, p}(k))_{k \\geq 0}$ for fixed $n$ and discuss the unimodality and log-concavity of such sequences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-13T15:59:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15",
      "05A19"
    ],
    "keywords": [
      "binary words",
      "subsequence frequency",
      "first set",
      "explicit expressions",
      "small values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}