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Home > Vol 28, No 2 (2008)

Wenhua Zhao

Illinois State University Department of Mathematics Normal IL 61790-4520 USA

In this paper, we construct explicitly a noncommutative symmetric ( N {\mathcal{N}} CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the N \mathcal {N} CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed N \mathcal {N} CS system with the order polynomials of rooted trees is also given and proved.

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