  Using the description of hypermaps with matchings, Goulden and Jackson have
given an expression of the generating series of rooted bipartite maps in terms
of the zonal polynomials. We generalize this approach to the case of
constellations on non-oriented surfaces that have recently been introduced by
Chapuy and Do{\l}\k{e}ga. A key step in the proof is an encoding of
constellations with tuples of matchings.
  We consider a one parameter deformation of the generating series of
constellations using Jack polynomials and we introduce the coefficients
$c^\lambda_{\mu^0,...,\mu^k}(b)$ obtained by the expansion of these functions
in the power-sum basis. These coefficients are indexed by $k+2$ integer
partitions and the deformation parameter $b$, and can be considered as a
generalization for $k\geq1$ of the connection coefficients introduced by
Goulden and Jackson. We prove that when we take some marginal sums, these
coefficients enumerate $b$-weighted $k$-tuples of matchings. This can be seen
as an "unrooted" version of a recent result of Chapuy and Do{\l}\k{e}ga for
constellations. For $k=1$, this gives a partial answer to Goulden and Jackson
Matching-Jack conjecture.
  Lassale has formulated a positivity conjecture for the coefficients
$\theta^{(\alpha)}_\mu(\lambda)$, defined as the coefficient of the Jack
polynomial $J_\lambda^{(\alpha)}$ in the power-sum basis. We use the second
main result of this paper to give a proof of this conjecture in the case of
partitions $\lambda$ with rectangular shape.
