On Jack’s connection coefficients and their computation

The class algebra and the double coset algebra are two commutative subalgebras of the group algebra of the symmetric group. The connection coefficients of these two algebraic structures received significant attention in combinatorics as they provide the number of factorizations of a given permutation into an ordered product of permutations satisfying given cyclic structures. While they are usually studied separately, these two families of connection coefficients share strong similarities. They are both equal to some sums of characters, respectively the irreducible characters of the symmetric group and the zonal spherical functions, two specific cases of a more general family of characters named Jack's characters.
Jack's characters are defined as the coefficients in the power sum expansion of the Jack's symmetric functions, a family of symmetric polynomials indexed by a parameter $\alpha$. Connection coefficients of the class algebra corresponds to the case $\alpha =1$ (Jack's symmetric functions are proportional to Schur polynomials in this case) and the connection coefficients of the double coset algebra corresponds to the case $\alpha = 2$ (Jack's symmetric functions are equal to zonal polynomials). We define Jack's connection coefficients to provide a unified approach for general parameter $\alpha$. This paper introduces these generalized coefficients and focus on their computations. More specifically we focus on the generalization of the formula giving the number of factorizations of a permutation of a given cyclic structure into the product of $r$ transpositions. We use the action of the Laplace–Beltrami operator on Jack's symmetric functions to provide a general formula and make this formula explicit for some given values of $r$.