A Ncs System over the Grossman-larson Hopf Algebra of Labeled Rooted Trees

In this paper, we construct explicitly a NCS system ([Z4]) Ω T ∈ (H GL) ×5 over the Grossman-Larson Hopf algebra H GL ([GL] and [F]) of rooted trees labeled by elements of a nonempty W ⊆ N of positive integers. By the universal property of the NCS system (NSym,Π) formed by the generating functions of certain NCSF’s ([GKLLRT]), we obtain a graded Hopf algebra homomorphism TW : NSym → H GL such that T (Π) = Ω T . Consequently, we get a specialization of NCSF’s by W -labeled tree and a graded Hopf algebra homomorphism T W : H W CK → QSym from the Connes-Kreimer Hopf algebra H CK of W -labeled rooted forests to the Hopf algebra QSym of Quasi-symmetric functions ([Ge], [MR] and [St2]). A connection of the coefficients of the third generating function of the NCS system Ω T with the order polynomials of rooted trees is also given and proved.