{
  "id": "math/0507169",
  "version": "v2",
  "published": "2005-07-08T06:46:45.000Z",
  "updated": "2005-07-20T21:43:56.000Z",
  "title": "A combinatorial interpretation of the eigensequence for composition",
  "authors": [
    "David Callan"
  ],
  "comment": "minor corrections and improvements",
  "categories": [
    "math.CO"
  ],
  "abstract": "The monic sequence that shifts left under convolution with itself is the Catalan numbers with 130+ combinatorial interpretations. Here we establish a combinatorial interpretation for the monic sequence that shifts left under composition: it counts permutations that contain a 3241 pattern only as part of a 35241 pattern. We give two recurrences, the first allowing relatively fast computation, the second similar to one for the Catalan numbers. Among the 4 times 4! = 96 similarly restricted patterns involving 4 letters (such as 4\\underline{2}31: a 431 pattern only occurs as part of a 4231), four different counting sequences arise: 64 give the Catalan numbers, 16 give the Bell numbers, 12 give sequence A051295 in OEIS, and 4 give a new sequence with an explicit formula.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-07-20T21:43:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15"
    ],
    "keywords": [
      "combinatorial interpretation",
      "catalan numbers",
      "composition",
      "monic sequence",
      "shifts left"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......7169C"
    }
  }
}