{
  "id": "1112.4970",
  "version": "v2",
  "published": "2011-12-21T10:00:51.000Z",
  "updated": "2011-12-22T07:41:21.000Z",
  "title": "Bijections and symmetries for the factorizations of the long cycle",
  "authors": [
    "Olivier Bernardi",
    "Alejandro H. Morales"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the factorizations of the permutation $(1,2,...,n)$ into $k$ factors of given cycle types. Using representation theory, Jackson obtained for each $k$ an elegant formula for counting these factorizations according to the number of cycles of each factor. In the cases $k=2,3$ Schaeffer and Vassilieva gave a combinatorial proof of Jackson's formula, and Morales and Vassilieva obtained more refined formulas exhibiting a surprising symmetry property. These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this article to establish a series of bijections which unveil some of the combinatorial properties of the factorizations of $(1,2,...,n)$ into $k$ factors for all $k$. We thereby obtain refinements of Jackson's formulas which extend the cases $k=2,3$ treated by Morales and Vassilieva. Our bijections are described in terms of \"constellations\", which are graphs embedded in surfaces encoding the transitive factorizations of permutations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-12-22T07:41:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "long cycle",
      "factorizations",
      "bijections",
      "jacksons formula",
      "rich combinatorial theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.4970B"
    }
  }
}