  This paper is devoted to the explicit computation of generating series for
the connection coefficients of two commutative subalgebras of the group algebra
of the symmetric group, the class algebra and the double coset algebra. As
shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral
distribution of some random matrices that are of interest to statisticians.
Morales and Vassilieva (2009, 2011) found explicit formulas for these
generating series in terms of monomial symmetric functions by introducing a
bijection between partitioned hypermaps on (locally) orientable surfaces and
some decorated forests and trees. Thanks to purely algebraic means, we recover
the formula for the class algebra and provide a new simpler formula for the
double coset algebra. As a salient ingredient, we derive a new explicit
expression for zonal polynomials indexed by partitions of type [a,b,1^(n-a-b)].
