  Introduced by Goulden and Jackson in their 1996 paper, the matchings-Jack
conjecture and the hypermap-Jack conjecture (also known as the $b$-conjecture)
are two major open questions relating Jack symmetric functions, the
representation theory of the symmetric groups and combinatorial maps. They show
that the coefficients in the power sum expansion of some Cauchy sum for Jack
symmetric functions and in the logarithm of the same sum interpolate
respectively between the structure constants of the class algebra and the
double coset algebra of the symmetric group and between the numbers of
orientable and locally orientable hypermaps. They further provide some evidence
that these two families of coefficients indexed by three partitions of a given
integer $n$ and the Jack parameter $\alpha$ are polynomials in $\beta =
\alpha-1$ with non negative integer coefficients of combinatorial significance.
This paper is devoted to the case when one of the three partitions is equal to
$(n)$. We exhibit some polynomial properties of both families of coefficients
and prove a variation of the hypermap-Jack conjecture and the matchings-Jack
conjecture involving labelled hypermaps and matchings in some important cases.
