A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

Authors:
Wenhua Zhao
Affiliations:
Department of Mathematics, Illinois State University, Normal, USA 61790-4520
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2008

Citing 2
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Enumerative combinatorics

Enumerative combinatorics
Noncommutative Symmetric Functions Iv: Quantum Linear Groups andHecke Algebras at {\bi q}\,{\bf =}\,{\bf 0}

Journal of Algebraic Combinatorics: An International Journal

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Abstract

In this paper, we construct explicitly a noncommutative symmetric ( ${\mathcal{N}}$ CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the $\mathcal {N}$ 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 $\mathcal {N}$ CS system with the order polynomials of rooted trees is also given and proved.