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

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 α . \alpha . We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in b b , the Jack parameter shifted by 1 1 . More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counting series in b 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 1 and 2 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 b , and we make the Hypermap-Jack Conjecture, that the coefficients are counting series in b b for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.