{
  "id": "0810.4901",
  "version": "v1",
  "published": "2008-10-27T18:15:03.000Z",
  "updated": "2008-10-27T18:15:03.000Z",
  "title": "Klazar trees and perfect matchings",
  "authors": [
    "David Callan"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Martin Klazar computed the total weight of ordered trees under 12 different notions of weight. The last and perhaps most interesting of these weights, w_{12}, led to a recurrence relation and an identity for which he requested combinatorial explanations. Here we provide such explanations. To do so, we introduce the notion of a \"Klazar violator\" vertex in an increasing ordered tree and observe that w_{12} counts what we call Klazar trees--increasing ordered trees with no Klazar violators. A highlight of the paper is a bijection from n-edge increasing ordered trees to perfect matchings of [2n]={1,2,...,2n} that sends Klazar violators to even numbers matched to a larger odd number. We find the distribution of the latter matches and, in particular, establish the one-summation explicit formula sum_{k=1}^{lfloor n/2 rfloor}(2k-1)!!^2 StirlingPartition{n+1}{2k+1} for the number of perfect matchings of [2n] with no even-to-larger-odd matches. The proofs are mostly bijective.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-10-27T18:15:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15"
    ],
    "keywords": [
      "perfect matchings",
      "klazar trees",
      "sends klazar violators",
      "larger odd number",
      "one-summation explicit formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.4901C"
    }
  }
}