{
  "id": "1109.2661",
  "version": "v1",
  "published": "2011-09-13T01:48:54.000Z",
  "updated": "2011-09-13T01:48:54.000Z",
  "title": "Counting Humps in Motzkin paths",
  "authors": [
    "Yun Ding",
    "Rosena R. X. Du"
  ],
  "comment": "8 pages, 2 Figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\\\"{o}der paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order $n$ with $k$ peaks is the Narayana number. By double counting super Schr\\\"{o}der paths, we also get an identity involving products of binomial coefficients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-09-13T01:48:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15"
    ],
    "keywords": [
      "motzkin paths",
      "counting humps",
      "super dyck paths",
      "similar relation",
      "narayana number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.2661D"
    }
  }
}