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

Explicit monomial expansions of the generating series for connection coefficients