{
  "id": "2004.03122",
  "version": "v1",
  "published": "2020-04-07T04:35:13.000Z",
  "updated": "2020-04-07T04:35:13.000Z",
  "title": "A new rank of partitions with overline designated summands",
  "authors": [
    "Robert. X. J. Hao",
    "Erin Y. Y. Shen",
    "Wenston J. T. Zang"
  ],
  "comment": "21 pages, 0 figure",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Andrews, Lewis and Lovejoy introduced the partition function $PD(n)$ as the number of partitions of $n$ with designated summands. In a recent work, Lin studied a partition function $PD_{t}(n)$ which counts the number of tagged parts over all the partitions of $n$ with designated summands. He proved that $PD_{t}(3n+2)$ is divisible by $3$. In this paper, we first introduce the structure of partitions with overline designated summands, which is counted by $PD_t(n)$. We then define a generalized rank named $pdt$-rank on partitions with overline designated summands and provide a combinatorial interpretation for the congruence $PD_t(3n+2)\\equiv 0\\pmod{3}$. Let $N_{dt}(m,n)$ denote the number of partitions of $n$ with overline designated summands with $pdt$-rank $m$. We prove that $N_{dt}(m,n)\\leq N_{dt}(m,n+1)$ with integer $m\\not=0$, $n\\geq 1$ as well as $N_{dt}(m,n)\\leq N_{dt}(m+1,n)$ for $m\\geq 2$, $n\\geq 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-07T04:35:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "05A20",
      "11P83"
    ],
    "keywords": [
      "overline designated summands",
      "partition function",
      "combinatorial interpretation",
      "tagged parts",
      "generalized rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}