Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions

Goulden, I.; Jackson, D.

doi:10.1090/S0002-9947-96-01503-6

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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