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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions
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by I. P. Goulden and D. M. Jackson PDF
Trans. Amer. Math. Soc. 348 (1996), 873-892 Request permission

Abstract:

A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter $\alpha .$ We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in $b$, the Jack parameter shifted by $1$. More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counting series in $b$ for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families. The coefficients of a second series, essentially the logarithm of the first, specialize at values $1$ and $2$ of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in $b$, and we make the Hypermap-Jack Conjecture, that the coefficients are counting series in $b$ for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.
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Additional Information
  • I. P. Goulden
  • Affiliation: Department Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
  • MR Author ID: 75735
  • Email: ipgoulden@math.uwaterloo.ca
  • D. M. Jackson
  • Affiliation: Department Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
  • MR Author ID: 92555
  • Email: dmjackson@watdragon.uwaterloo.ca
  • Received by editor(s): November 27, 1994
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 873-892
  • MSC (1991): Primary 05E05, 05A15, 57M15
  • DOI: https://doi.org/10.1090/S0002-9947-96-01503-6
  • MathSciNet review: 1325917