{
  "id": "1601.01501",
  "version": "v1",
  "published": "2016-01-07T12:01:11.000Z",
  "updated": "2016-01-07T12:01:11.000Z",
  "title": "Cumulants of Jack symmetric functions and $b$-conjecture",
  "authors": [
    "Maciej Dołęga",
    "Valentin Féray"
  ],
  "comment": "26 pages, 2 figures, comments are welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\\psi(x, y, z; t, 1+\\beta)$ that might be interpreted as a continuous deformation of generating series of rooted hypermaps. They made the following conjecture: the coefficients of $\\psi(x, y, z; t, 1+\\beta)$ in the power-sum basis are polynomials in $\\beta$ with nonnegative integer coefficients (by construction, these coefficients are rational functions in $\\beta$). We prove partially this conjecture, nowadays called $b$-conjecture, by showing that coefficients of $\\psi(x, y, z; t, 1+ \\beta)$ are polynomials in $\\beta$ with rational coefficients. A key step of the proof is a strong factorization property of Jack polynomials when $\\alpha$ tends to $0$, that may be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-07T12:01:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05"
    ],
    "keywords": [
      "jack symmetric functions",
      "conjecture",
      "strong factorization property",
      "multivariate generating series",
      "independent interest"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160101501D"
    }
  }
}