Connection coefficients, matchings, maps and

combinatorial conjectures for

Jack symmetric functions

Authors:
I. P. Goulden and D. M. Jackson

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

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Abstract | References | Similar Articles | Additional Information

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 We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in , the Jack parameter shifted by . More strongly, we make the * Matchings-Jack Conjecture*, that the coefficients are counting series in 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 and 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 , and we make the *Hypermap-Jack Conjecture*, that the coefficients are counting series in 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

Email:
ipgoulden@math.uwaterloo.ca

**D. M. Jackson**

Affiliation:
Department Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Email:
dmjackson@watdragon.uwaterloo.ca

DOI:
https://doi.org/10.1090/S0002-9947-96-01503-6

Received by editor(s):
November 27, 1994

Article copyright:
© Copyright 1996
American Mathematical Society