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We introduce analogs of the Hopf algebra of Free quasi-symmetric functionswith bases labelled by colored permutations. When the color set is a semigroup,an internal product can be introduced. This leads to the construction ofgeneralized descent algebras associated with wreath products $\Gamma\wr\SG_n$and to the corresponding generalizations of quasi-symmetric functions. Theassociated Hopf algebras appear as natural analogs of McMahon's multisymmetricfunctions. As a consequence, we obtain an internal product on ordinarymulti-symmetric functions. We extend these constructions to Hopf algebras ofcolored parking functions, colored non-crossing partitions and parkingfunctions of type B.

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions