{
  "id": "2209.01639",
  "version": "v1",
  "published": "2022-09-04T15:22:12.000Z",
  "updated": "2022-09-04T15:22:12.000Z",
  "title": "Arithmetic properties of certain $t$-regular partitions",
  "authors": [
    "Rupam Barman",
    "Ajit Singh",
    "Gurinder Singh"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a positive integer $t\\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$. We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of $b_9(n)$ and $b_{19}(n)$ modulo $2$. We also relate $b_{t}(n)$ to the ordinary partition function, and prove that $b_{t}(n)$ satisfies the Ramanujan's famous congruences for some infinite families of $t$. For $t\\in \\{6,10,14,15,18,20,22,26,27,28\\}$, Keith and Zanello conjectured that there are no integers $A>0$ and $B\\geq 0$ for which $b_t(An+ B)\\equiv 0\\pmod 2$ for all $n\\geq 0$. We prove that, for any $t\\geq 2$ and prime $\\ell$, there are infinitely many arithmetic progressions $An+B$ for which $\\sum_{n=0}^{\\infty}b_t(An+B)q^n\\not\\equiv0 \\pmod{\\ell}$. Next, we obtain quantitative estimates for the distributions of $b_{6}(n), b_{10}(n)$ and $b_{14}(n)$ modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and $13$-regular partition functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-04T15:22:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic properties",
      "ordinary partition function",
      "regular partition functions",
      "establish infinite families",
      "congruences modulo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}