{
  "id": "1706.07148",
  "version": "v1",
  "published": "2017-06-22T02:12:52.000Z",
  "updated": "2017-06-22T02:12:52.000Z",
  "title": "On the Enumeration and Congruences for m-ary Partitions",
  "authors": [
    "Lisa Hui Sun",
    "Mingzhi Zhang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $m\\ge 2$ be a fixed positive integer. Suppose that $m^j \\leq n< m^{j+1}$ is a positive integer for some $j\\ge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this paper, we show that $b_m(n)$ can be represented as a $j$-fold summation by constructing a one-to-one correspondence between the $m$-ary partitions and a special class of integer sequences rely only on the base $m$ representation of $n$. It directly reduces to Andrews, Fraenkel and Sellers' characterization of the values $b_{m}(mn)$ modulo $m$. Moreover, denote $c_{m}(n)$ the number of $m$-ary partitions of $n$ without gaps, wherein if $m^i$ is the largest part, then $m^k$ for each $0\\leq k<i$ also appears as a part. We also obtain an enumeration formula for $c_m(n)$ which leads to an alternative representation for the congruences of $c_m(mn)$ due to Andrews, Fraenkel, and Sellers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-22T02:12:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "m-ary partitions",
      "congruences",
      "positive integer",
      "enumeration formula",
      "one-to-one correspondence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}