{
  "id": "1111.1489",
  "version": "v2",
  "published": "2011-11-07T05:11:54.000Z",
  "updated": "2012-08-22T02:22:23.000Z",
  "title": "Euler's Partition Theorem with Upper Bounds on Multiplicities",
  "authors": [
    "William Y. C. Chen",
    "Ae Ja Yee",
    "Albert J. W. Zhu"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We show that the number of partitions of n with alternating sum k such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For m=0, our result reduces to Bessenrodt's refinement of Euler's Theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a connection to a generalization of Euler's theorem, which can be deduced from a theorem of Andrews on equivalent upper bound sequences of multiplicities. Analogously, we show that the number of partitions of n with alternating sum k such that the multiplicity of each even part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is also bounded by 2m+1. We provide a combinatorial proof as well.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-08-22T02:22:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P81"
    ],
    "keywords": [
      "eulers partition theorem",
      "multiplicity",
      "odd parts",
      "alternating sum",
      "eulers theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.1489C"
    }
  }
}