## 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