Connection Coefficients, Matchings, Maps and Combinatorial Conjectures for Jack Symmetric Functions

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. 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.