Polynomial properties of Jack connection coefficients and generalization of a result by Dénes

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. These two families of coefficients are of significant interest in the theory of symmetric functions as well as in the combinatorics of factorizations of permutations. While they share similar properties, they are usually studied separately. First (partially) introduced by Goulden and Jackson (Trans Am Math Soc 348(3):873–892, 1996), Jack connection coefficients provide a natural unified approach closely related to the theory of Jack polynomials. Goulden and Jackson conjectured that these coefficients are polynomials in the Jack parameter $$\alpha $$α with nice combinatorial properties, the 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 formulas including notably a generalization of a classical formula of Dénes for the number of minimal factorizations of a permutation into transpositions.