Hopf Algebra Methods In Combinatorial Mathematics

As Hopf Algebra theory getting more and more perfectly, it is not any long an isolated system, it made close relations with many other territories in mathematics. Its applications get more and more widely which in graph theory, mathematics-physics, discrete-mathematics, etc. Combinatorial Hopf Algebra, which is a new theory developped in recent years, become an object that is being paied close attention to by algebra scholars and combintorial scholars together.In 1979, Joni and Rota [12] made the foundamental observation that many discrete structures give rise to natural Hopf algebras whose comultiplications encode the disassembly of those structures. This was further developed by Schmitt [24] . A first link between these Hopf algebras and quasi- symmetric functions was found by Ehrenborg [9] , who encoded the flag vetor of a graded poset as a morphism from a Hopf algebra of graded posets to QSym . A similar construction was made by Bergeron and Sottile given [4], They have shown that a quasi-symmetric function associated to an edge-labeled poset also gives a morphism of Hopf algebras. In 2003, M. Aguiar, N. Bergeron and F. Sottile [1] made a new defination of Combinatorial Hopf Algebra. In this paper, we build and analyze several Combinatorial Hopf Algebras mainly, and made some conclusions.The thesis is composed of five chapters.Chapter One is the introduction of this paper, which introduces the problem and the background of the problem we have studied.Chapter Two is preparatory knowledge, which contains two sections, they are the basic notions of Combinatorial Hopf algebra and related knowledge...