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We define a new basis of the algebra of quasi-symmetric functions by liftingthe cycle-index polynomials of symmetric groups to noncommutative polynomialswith coefficients in the algebra of free quasi-symmetric functions, and thenprojecting the coefficients to $QSym$. By duality, we obtain a basis ofnoncommutative symmetric functions, for which a product formula and arecurrence in the form of a combinatorial complex are obtained. This basisallows to identify noncommutative symmetric functions with the quotient ofFQSym induced by the pattern-replacement relation $321 \equiv 231$ and $312\equiv 132$.

A noncommutative cycle index and new bases of quasi-symmetric functions and noncommutative symmetric functions