A Noncommutative Cycle Index and New Bases of Quasi-symmetric Functions and Noncommutative Symmetric Functions


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 \(\mathrm{FQSym}\) induced by the pattern-replacement relation \(321\equiv 231\) and \(312\equiv 132\).

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Correspondence to Jean-Yves Thibon.

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Novelli, J., Thibon, J. & Toumazet, F. A Noncommutative Cycle Index and New Bases of Quasi-symmetric Functions and Noncommutative Symmetric Functions. Ann. Comb. (2020). https://doi.org/10.1007/s00026-020-00504-5

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  • Noncommutative symmetric functions
  • Quasi-symmetric functions
  • Dendriform algebras

Mathematics Subject Classification

  • 16T30
  • 05E05
  • 05A18