{
  "id": "2211.15277",
  "version": "v1",
  "published": "2022-11-28T13:05:59.000Z",
  "updated": "2022-11-28T13:05:59.000Z",
  "title": "$q$-enumeration of type B and D Eulerian polynomials based on parity of descents",
  "authors": [
    "Hiranya Kishore Dey",
    "Umesh Shankar",
    "Sivaramakrishnan Sivasubramanian"
  ],
  "comment": "21 pages. Comments are welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "Carlitz and Scoville in 1973 considered four-variable polynomials enumerating the permutations according to the parity of both descents and ascents. In a recent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville's generating function by counting the inversion number. Moreover, they also proved a type B analogue by enumerating the signed permutations with respect to the parity of descent and ascent position. In this work we prove a $q$-analogue of the type B result of Pan and Zeng by counting the type $B$ inversion number. We also obtain a $q$-analogue of the generating functions for the bivariate alternating descent polynomials. Similar results are also obtained for type D Coxeter groups. As a by-product of our proofs, we get $q$-analogues of Hyatt's recurrences for the type B Eulerian polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-28T13:05:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "05A15",
      "05E16"
    ],
    "keywords": [
      "eulerian polynomials",
      "inversion number",
      "enumeration",
      "bivariate alternating descent polynomials",
      "permutations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}