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A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions. (English) Zbl 1460.16037
Summary: We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to QSym. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of $$\text{FQSym}$$ induced by the pattern-replacement relation $$321\equiv 231$$ and $$312\equiv 132$$.
##### MSC:
 16T30 Connections of Hopf algebras with combinatorics 05E05 Symmetric functions and generalizations 05A18 Partitions of sets
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##### References:
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