{
  "id": "2212.13555",
  "version": "v1",
  "published": "2022-12-27T17:16:43.000Z",
  "updated": "2022-12-27T17:16:43.000Z",
  "title": "Systematic study of Schmidt-type partitions via weighted words",
  "authors": [
    "Isaac Konan"
  ],
  "comment": "17 pages. 2 figures",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $S$ be a set of positive integers. In this paper, we provide an explicit formula for $$\\sum_{\\la} C(\\la) q^{\\sum_{i\\in S} \\la_i}$$ where $\\la=(\\la_1,\\ldots)$ run through some subsets of over-partitions, and $C(\\la)$ is a certain product of ``colors'' assigned to the parts of $\\la$. This formula allows us not only to retrieve several known Schmidt-type theorems but also to provide new Schmidt-type theorems for sets $S$ with non-periodic gaps. The example of $S=\\{n(n-1)/2+1:n\\in 1\\}$ leads to the following statement: for all non-negative integer $m$, the number of partitions such that $\\sum_{i\\in S}\\la_i =m$ is equal to the number of plane partitions of $m$. Furthermore, we introduce a new family of partitions, the block partitions, generalizing the $k$-elongated partitions. From that family of partitions, we provide a generalization of a Schmidt-type theorem due to Andrews and Paule regarding $k$-elongated partitions and establish a link with the Eulerian polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-27T17:16:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83",
      "05A15"
    ],
    "keywords": [
      "systematic study",
      "schmidt-type partitions",
      "weighted words",
      "schmidt-type theorem",
      "elongated partitions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}