Date of Submission
This project is concerned with the use of Hopf algebras to study combinatorial questions about graphs and posets. We discuss Stanley's connection between the chromatic polynomial of a graph G and the acyclic orientations of G. We then recast Stanley's result using the language of Hopf algebras, and this correspondence is described in terms of inverting a character. Applying this Hopf algebraic machinery to posets, we explore a character-theoretic approach to studying the enumeration of order-preserving maps. We use this machinery to derive an algebraic proof of a reciprocity theorem relating strict to weak order-preserving maps. Moreover, we give a new proof of Stanley's graph theory result. Finally, we create a new Hopf algebra of "acyclically oriented graphs," and use it to summarize the connection between graphs and posets.
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Selfridge, Benjamin G., "Characters of Combinatorial Hopf Algebras" (2011). Senior Projects Spring 2011. 187.