{
  "id": "1601.01595",
  "version": "v1",
  "published": "2016-01-07T16:50:33.000Z",
  "updated": "2016-01-07T16:50:33.000Z",
  "title": "Compositions colored by simplicial polytopic numbers",
  "authors": [
    "Daniel Birmajer",
    "Juan B. Gil",
    "Michael D. Weiner"
  ],
  "comment": "9 pages. Submitted for publication",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "For a given integer $d\\ge 1$, we consider $\\binom{n+d-1}{d}$-color compositions of a positive integer $\\nu$ for which each part of size $n$ admits $\\binom{n+d-1}{d}$ colors. We give explicit formulas for the enumeration of such compositions, generalizing existing results for $n$-color compositions (case $d=1$) and $\\binom{n+1}{2}$-color compositions (case $d=2$). In addition, we give bijections from the set of $\\binom{n+d-1}{d}$-color compositions of $\\nu$ to the set of compositions of $(d+1)\\nu - 1$ having only parts of size $1$ and $d+1$, the set of compositions of $(d+1)\\nu$ having only parts of size congruent to $1$ modulo $d+1$, and the set of compositions of $(d+1)\\nu + d$ having no parts of size less than $d+1$. Our results rely on basic properties of partial Bell polynomials and on a suitable adaptation of known bijections for $n$-color compositions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-07T16:50:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "11B75"
    ],
    "keywords": [
      "simplicial polytopic numbers",
      "color compositions",
      "partial bell polynomials",
      "bijections",
      "basic properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}