{
  "id": "1909.13549",
  "version": "v1",
  "published": "2019-09-30T09:28:32.000Z",
  "updated": "2019-09-30T09:28:32.000Z",
  "title": "Partitions into polynomial with the number of parts in arithmetic progression",
  "authors": [
    "Nian Hong Zhou"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $k, a\\in \\mathbb{N}$ and let $f(x)\\in\\mathbb{Q}[x]$ be a polynomial with degree ${\\rm deg}(f)\\ge 1$ such that all elements of the sequence $\\{f(n)\\}_{n\\in\\mathbb{N}}$ lies on $ \\mathbb{N}$ and $\\gcd(\\{f(n)\\}_{n\\in\\mathbb{N}})=1$. Let $p_f(n)$ and $p_f(a,k;n)$ denotes the number of partitions of integer $n$ whose parts taken from the sequence $\\{f(n)\\}_{n\\in\\mathbb{N}}$ and the number of parts of those partitions are congruent to $a$ modulo $k$, respectively. We prove that there exist a constant $\\delta_{f}\\in\\mathbb{R}_+$ depending only on $f$ such that $$p_f(a,k;n)=\\frac{p_f(n)}{k}\\left[1+O\\left(n\\exp\\left(-\\delta_{f}k^{-2}n^{\\frac{1}{1+{\\rm deg}(f)}}\\right)\\right)\\right],$$ holds uniformly for all $a,k, n\\in\\mathbb{N}$ with $k^{2+2{\\rm deg}(f)}\\ll n$, as $n$ tends to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-30T09:28:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P82",
      "11P83",
      "05A17",
      "11L07"
    ],
    "keywords": [
      "arithmetic progression",
      "partitions",
      "polynomial",
      "parts taken"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}