{
  "id": "1407.5433",
  "version": "v2",
  "published": "2014-07-21T09:35:12.000Z",
  "updated": "2014-10-31T12:23:33.000Z",
  "title": "Arithmetic Properties of Partition Triples With Odd Parts Distinct",
  "authors": [
    "Liuquan Wang"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathrm{pod}_{-3}(n)$ denote the number of partition triples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\\mathrm{pod}_{-3}(n)$ involving the following infinite family of congruences: for any integers $\\alpha \\ge 1$ and $n\\ge 0$, \\[\\mathrm{pod}_{-3}\\Big({{3}^{2\\alpha +2}}n+\\frac{23\\times {{3}^{2\\alpha +1}}+3}{8}\\Big)\\equiv 0 \\pmod{9}.\\] We also establish some arithmetic relations between $\\mathrm{pod}(n)$ and $\\mathrm{pod}_{-3}(n)$, as well as some congruences for $\\mathrm{pod}_{-3}(n)$ modulo 7 and 11.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-21T09:35:12.000Z",
      "abstract": "Let $\\mathrm{pod}_{-3}(n)$ denote the number of partition triples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\\mathrm{pod}_{-3}(n)$ involving the following infinite family of congruences: for any integers $\\alpha \\ge 1$ and $n\\ge 0$, \\[\\mathrm{pod}_{-3}\\Big({{3}^{2\\alpha +2}}n+\\frac{23\\times {{3}^{2\\alpha +1}}+3}{8}\\Big)\\equiv 0\\,(\\bmod \\, 9).\\] Some arithmetic relations between $\\mathrm{pod}(n)$ and $\\mathrm{pod}_{-3}(n)$, as well as some congruences modulo 7 and 11 for $\\mathrm{pod}_{-3}(n)$ have also been established.",
      "comment": "This is an original research article about partitions. It's first completed by the author in Janunary 2014, and submitted for publication in April",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-31T12:23:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "odd parts distinct",
      "partition triples",
      "arithmetic properties",
      "congruences modulo",
      "arithmetic relations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5433W"
    }
  }
}