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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 1325917
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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
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: http://dx.doi.org/10.1090/S0002-9947-96-01503-6
PII: S 0002-9947(96)01503-6
Received by editor(s): November 27, 1994
Article copyright: © Copyright 1996 American Mathematical Society