{
  "id": "2005.03619",
  "version": "v1",
  "published": "2020-05-07T17:18:21.000Z",
  "updated": "2020-05-07T17:18:21.000Z",
  "title": "On the partitions into distinct parts and odd parts",
  "authors": [
    "Mircea Merca"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we show that the difference between the number of parts in the odd partitions of $n$ and the number of parts in the distinct partitions of $n$ satisfies Euler's recurrence relation for the partition function $p(n)$ when $n$ is odd. A decomposition of this difference in terms of the total number of parts in all the partitions of $n$ is also derived. In this context, we conjecture that for $k>0$, the series $$ (q^2;q^2)_\\infty \\sum_{n=k}^\\infty \\frac{q^{{k\\choose 2}+(k+1)n}}{(q;q)_n} \\begin{bmatrix} n-1\\\\k-1 \\end{bmatrix} $$ has non-negative coefficients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-07T17:18:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P81",
      "05A17"
    ],
    "keywords": [
      "odd parts",
      "distinct parts",
      "satisfies eulers recurrence relation",
      "partition function",
      "distinct partitions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}