  This article is devoted to the computation of Jack connection coefficients, a
generalization of the connection coefficients of two classical commutative
subalgebras of the group algebra of the symmetric group: the class algebra and
the double coset algebra. The connection coefficients of these two algebraic
structures are of significant interest in the study of Schur and zonal
polynomials as well as the irreducible characters of the symmetric group and
the zonal spherical functions. Furthermore they play an important role in
combinatorics as they give the number of factorizations of a permutation into a
product of permutations with given cyclic properties. Usually studied
separately, these two families of coefficients share strong similar properties.
First (partially) introduced by Goulden and Jackson in 1996, Jack connection
coefficients provide a natural unified approach closely related to the theory
of Jack polynomials, a family of bases in the ring of symmetric functions
indexed by a parameter \alpha that generalizes both Schur (case \alpha = 1) and
zonal polynomials (case \alpha = 2). Jack connection coefficients are also
directly linked to Jack characters, a general view of the characters of the
symmetric group and the zonal spherical functions. Goulden and Jackson
conjectured that these coefficients are polynomials in \alpha with nice
combinatorial properties, the so-called Matchings-Jack conjecture. In this
paper, we use the theory of Jack symmetric functions and the Laplace Beltrami
operator to show the polynomial properties of Jack connection coefficients in
some important cases. We also provide explicit formulations including notably a
generalization of a classical formula of D\'enes for the number of minimal
factorizations of a permutation into transpositions.
