{
  "id": "1604.07897",
  "version": "v1",
  "published": "2016-04-27T01:03:45.000Z",
  "updated": "2016-04-27T01:03:45.000Z",
  "title": "Some aspects of (r,k)-parking functions",
  "authors": [
    "Richard Stanley",
    "Yinghui Wang"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "An (r,k)-parking function} of length n may be defined as a sequence (a_1,...,a_n) of positive integers whose increasing rearrangement b_1\\leq ... \\leq b_n satisfies b_i\\leq k+(i-1)r. The case r=k=1 corresponds to ordinary parking functions. We develop numerous properties of (r,k)-parking functions. In particular, if F_n^{(r,k)} denotes the Frobenius characteristic of the action of the symmetric group S_n on the set of all (r,k)-parking functions of length n, then we find a combinatorial interpretation of the coefficients of the power series (\\sum_{n\\geq 0}F_n^{(r,1)}t^n)^k for any integer k. For instance, when k>0 this power series is just \\sum_{n\\geq 0} F_n^{(r,k)} t^n. We also give a q-analogue of this result. For fixed r, we can use the symmetric functions F_n^{(r,1)} to define a multiplicative basis for the ring of symmetric functions. We investigate some of the properties of this basis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-27T01:03:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05"
    ],
    "keywords": [
      "symmetric functions",
      "power series",
      "combinatorial interpretation",
      "symmetric group",
      "ordinary parking functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160407897S"
    }
  }
}