2018/04/05 by Novelli, Jean-Christophe, Thibon, Jean-Yves, Toumazet, Frederic
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1804.01762
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 FQSym induced by the pattern-replacement relation 321 ≡ 231 and 312 ≡ 132.