The top-degree part in the Matchings-Jack Conjecture

TL;DR

This paper investigates the top-degree part of the coefficients in the Matchings-Jack Conjecture, providing conditions for polynomial degree bounds and revealing the positivity of the leading coefficient.

Contribution

It establishes a necessary and sufficient condition for the polynomial degree bound and characterizes the leading coefficient as a positive integer.

Findings

Provided a condition for the polynomial degree bound of coefficients.

Proved the positivity of the leading coefficient.

Connected the leading coefficient to the Matchings-Jack Conjecture.

Abstract

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