The Top-Degree Part in the Matchings-Jack Conjecture

In 1996 Goulden and Jackson introduced a family of coefficients $( c_{\pi, \sigma}^{\lambda} ) $ indexed by triples partitions which arise in the power sum expansion some Cauchy for Jack symmetric functions $(J^{(\alpha )}_\pi )$. The can be viewed as an interpolation between structure constants class algebra double coset algebra. suggested that are polynomials variable $\beta := \alpha-1$ with non-negative integer there is combinatorics matching hidden behind them. This Matchings-Jack Conjecture remains open. Do??ga Féray showed polynomiality connection $c^\lambda_{\pi,\sigma}$ gave upper bound on degrees. We show dual approach to this problem investigate characters their coefficients. give necessary sufficient condition polynomial \sigma}^{\lambda}$ achieve bound. leading coefficient positive we present it context Jackson.