{
  "id": "1409.8093",
  "version": "v1",
  "published": "2014-09-29T12:12:24.000Z",
  "updated": "2014-09-29T12:12:24.000Z",
  "title": "The sorting index on colored permutations and even-signed permutations",
  "authors": [
    "Sen-Peng Eu",
    "Yuan-Hsun Lo",
    "Tsai-Lien Wong"
  ],
  "comment": "27 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We define a new statistic $\\mathsf{sor}$ on the set of colored permutations $\\mathsf{G}_{r,n}$ and prove that it has the same distribution as the length function. For the set of restricted colored permutations corresponding to the arrangements of $n$ non-attacking rooks on a fixed Ferrers shape we show that the following two sequences of set-valued statistics are joint equidistributed: $(\\ell,\\mathsf{Rmil}^0,\\mathsf{Rmil}^1,...,\\mathsf{Rmil}^{r-1}$, $\\mathsf{Lmil}^0,\\mathsf{Lmil}^1,...,\\mathsf{Lmil}^{r-1}$, $\\mathsf{Lmal}^0,\\mathsf{Lmal}^1,...,\\mathsf{Lmal}^{r-1}$, $\\mathsf{Lmap}^0,\\mathsf{Lmap}^1,...,\\mathsf{Lmap}^{r-1})$ and $(\\mathsf{sor},\\mathsf{Cyc}^0,\\mathsf{Cyc}^{r-1},...,\\mathsf{Cyc}^{1}$, $\\mathsf{Lmic}^0,\\mathsf{Lmic}^{r-1},...,\\mathsf{Lmic}^{1}$, $\\mathsf{Lmal}^0,\\mathsf{Lmal}^1,...,\\mathsf{Lmal}^{r-1}$, $\\mathsf{Lmap}^0,\\mathsf{Lmap}^1,...,\\mathsf{Lmap}^{r-1})$. Analogous results are also obtained for Coxeter group of type $D$. Our results extend recent results of Petersen, Chen-Gong-Guo and Poznanovi\\'{c}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-29T12:12:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A19"
    ],
    "keywords": [
      "colored permutations",
      "even-signed permutations",
      "sorting index",
      "length function",
      "fixed ferrers shape"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.8093E"
    }
  }
}