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 $\alpha .$ 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

Article copyright:
© Copyright 1996
American Mathematical Society