{
  "id": "1412.0392",
  "version": "v1",
  "published": "2014-12-01T09:33:02.000Z",
  "updated": "2014-12-01T09:33:02.000Z",
  "title": "On the equation $\\mathbf{m=xyzw}$ with $\\mathbf{x\\leqslant y\\leqslant z\\leqslant w}$ in positive integers",
  "authors": [
    "Madjid Mirzavaziri",
    "Daniel Yaqubi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "As a well-known enumerative problem, the number of solutions of the equation $m=m_1+...+m_k$ with $m_1\\leqslant...\\leqslant m_k$ in positive integers is $\\Pi(m,k)=\\sum_{i=0}^k\\Pi(m-k,i)$ and $\\Pi$ is called the additive partition function. In this paper, we give a recursive formula for the so-called multiplicative partition function $\\mu_1(m,k):=$ the number of solutions of the equation $m=m_1... m_k$ with $m_1\\leqslant...\\leqslant m_k$ in positive integers. In particular, using an elementary proof, we give an explicit formula for the cases $k=1,2,3,4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-01T09:33:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive integers",
      "well-known enumerative problem",
      "additive partition function",
      "multiplicative partition function",
      "elementary proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.0392M"
    }
  }
}