A labelled variant of the matchings-Jack and hypermap-jack conjectures

Introduced by Goulden and Jackson in their 1996 paper, the matchings-Jack and hypermap-Jack conjectures are two major open questions relating symmetric functions, representation theory and combinatorial maps. These conjectures state an important combinatorial interpretation of the coefficients in the power sum expansion of two related formal power series involving Jack symmetric functions. These coefficients are indexed by three partitions of a given integer n. This paper is devoted to the case when one of them is equal to (n). We exhibit some of their polynomial properties and prove a variant of the two conjectures involving labelled hypermaps and matchings in some important cases.