{
  "id": "0911.5019",
  "version": "v1",
  "published": "2009-11-26T07:35:46.000Z",
  "updated": "2009-11-26T07:35:46.000Z",
  "title": "A Franklin Type Involution for Squares",
  "authors": [
    "William Y. C. Chen",
    "Eric H. Liu"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We find an involution as a combinatorial proof of a Ramanujan's partial theta identity. Based on this involution, we obtain a Franklin type involution for squares in the sense that the classical Franklin involution provides a combinatorial interpretation of Euler's pentagonal number theorem. This Franklin type involution can be considered as a solution to a problem proposed by Pak concerning the parity of the number of partitions of n into distinct parts with the smallest part being odd. Using a weighted form of our involution, we give a combinatorial proof of a weighted partition theorem derived by Alladi from Ramanujan's partial theta identity. This answers a question of Berndt, Kim and Yee. Furthermore, through a different weight assignment, we find combinatorial interpretations for another partition theorem derived by Alladi from a partial theta identity of Andrews. Moreover, we obtain a partition theorem based on Andrews' identity and provide a combinatorial proof by certain weight assignment for our involution. A specialization of our partition theorem is relate to an identity of Andrews concerning partitions into distinct nonnegative parts with the smallest part being even. Finally, we give a more general form of our partition theorem which in return corresponds to a generalization of Andrews' identity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-11-26T07:35:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P81"
    ],
    "keywords": [
      "franklin type involution",
      "ramanujans partial theta identity",
      "combinatorial proof",
      "smallest part",
      "eulers pentagonal number theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.5019C"
    }
  }
}