{
  "id": "1701.07915",
  "version": "v1",
  "published": "2017-01-27T00:53:21.000Z",
  "updated": "2017-01-27T00:53:21.000Z",
  "title": "An overpartition analogue of $q$-binomial coefficients, II: combinatorial proofs and $(q,t)$-log concavity",
  "authors": [
    "Jehanne Dousse",
    "Byungchan Kim"
  ],
  "comment": "19 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \\times n$ rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization $\\overline{{m+n \\brack n}}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of Delannoy numbers. First we obtain finite versions of classical $q$-series identities such as the $q$-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gaussian polynomials. Then, by constructing involutions, we obtain an identity involving a finite theta function and prove the $(q,t)$-log concavity of $\\overline{{m+n \\brack n}}_{q,t}$. We particularly emphasize the role of combinatorial proofs and the consequences of our results on Delannoy numbers. We conclude with some conjectures about the unimodality of $\\overline{{m+n \\brack n}}_{q,t}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-27T00:53:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P81",
      "11P84",
      "05A10",
      "05A17",
      "11B65",
      "05A20",
      "05A30"
    ],
    "keywords": [
      "log concavity",
      "overpartition analogue",
      "combinatorial proofs",
      "binomial coefficients",
      "gaussian polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}