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Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions. (English) Zbl 1231.05278
Summary: We introduce analogs of the Hopf algebra of free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products \(\Gamma\wr\mathfrak S_n\) and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type \(B\).

MSC:
05E05 Symmetric functions and generalizations
05A15 Exact enumeration problems, generating functions
05C05 Trees
16T30 Connections of Hopf algebras with combinatorics
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