The University of Michigan Combinatorics Seminar
Many generating functions of enumerative combinatorics are either symmetric or quasi-symmetric. Also, there are important eulerian objects, whose combinatorial invariants satisfy certain linear equations. The notion of a Combinatorial Hopf algebra attempts to explain this ubiquity of quasi-symmetric generating functions, while also giving this notion of eulerian an algebraic framework from which it can be generalized.
A Combinatorial Hopf algebra is a graded connected Hopf algebra H over a field F equipped with a multiplicative linear functional H --> F. The terminal object in the category of combinatorial Hopf algebras is the algebra of quasi-symmetric functions, which explains their ubiquity as generating functions in combinatorics. The Möobius function of a combinatorial Hopf algebra gives rise to natural Dehn-Sommerville relations on the Hopf algebra, and a combinatorial Hopf algebra satisfying these relations is eulerian. The relation of this theory to combinatorics is through Hopf algebras of combinatorial objects.
In this talk, we will define combinatorial Hopf algebras, and
show how the algebra of quasi-symmetric functions is a terminal
object in their category. We then define eulerian combinatorial
Hopf algebras and illustrate this theory with examples.