{
  "id": "2302.01253",
  "version": "v1",
  "published": "2023-02-02T17:38:40.000Z",
  "updated": "2023-02-02T17:38:40.000Z",
  "title": "$6$-regular partitions: new combinatorial properties, congruences, and linear inequalities",
  "authors": [
    "Cristina Ballantine",
    "Mircea Merca"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts not congruent to $\\pm 2$ modulo $6$, $Q_2(n)$, and investigate connections between $b_6(n)$ and $Q_2(n)$ providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler's partition function $p(n)$. Infinite families of linear inequalities involving the $6$-regular partition function $b_6(n)$ and the distinct partition function $Q_2(n)$ are proposed as open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-02T17:38:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P81",
      "11P82",
      "05A19",
      "05A20"
    ],
    "keywords": [
      "linear inequalities",
      "combinatorial properties",
      "infinite families",
      "eulers partition function",
      "regular partition function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}