{
  "id": "2301.00341",
  "version": "v1",
  "published": "2023-01-01T04:14:30.000Z",
  "updated": "2023-01-01T04:14:30.000Z",
  "title": "Symmetric polynomials connecting unsigned and signed relative derangements",
  "authors": [
    "Ricky Xiao-Feng Chen",
    "Yu-Chen Ruan"
  ],
  "comment": "Comments are all welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we introduce polynomials (in $t$) of signed relative derangements that track the number of signed elements. The polynomials are clearly seen to be in a sense symmetric. Note that relative derangements are those without any signed elements, i.e., the evaluations of the polynomials at $t=0$. Also, the numbers of all signed relative derangements are given by the evaluations at $t=1$. Then the coefficients of the polynomials connect unsigned and signed relative derangements and show how putting elements with signs affects the formation of derangements. We first prove a recursion satisfied by these polynomials which results in a recursion satisfied by the coefficients. A combinatorial proof of the latter is provided next. We also show that the sequences of the coefficients are unimodal. Moreover, other results are obtained. For instance, a kind of dual of a relation between signed derangements and signed relative derangements previously proved by Chen and Zhang is presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-01T04:14:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05A19",
      "05A15"
    ],
    "keywords": [
      "signed relative derangements",
      "symmetric polynomials connecting",
      "coefficients",
      "signed elements",
      "sense symmetric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}