Many enumeration problems in algebraic combinatorics lead to quasi-symmetric generating functions, for example Schur functions, P-partitions, and Stanley symmetric functions. Some of these have been shown to have interesting algebraic properties, most notably the Hopf structures of Ehrenborg's quasi-symmetric generating function for the flag f-vector of a polytope and of Bergeron and Sottile's generating function for decsents in a labeled poset.
This talk will describe joint work with Bergeron, Mykytiuk, and van Willigenburg which gives a unified construction of most such quasi-symmetric functions. Our approach is motivated by work on the Schubert calculus for flag manifolds and uses representations of the algebra of non-commutative symmetric functions generated by what we call Pieri operators.