In this talk, I will report on the following joint work with Nantel Bergeron of York University:
An edge-labeled poset is a graded poset, the edges of whose Hasse diagram are labeled with integers. While investigating the multiplication of Schubert polynomials, we were led to associate a symmetric function SP to every such poset satisfying a certain symmetry condition on its set of maximal chains. This gives a uniform construction of skew Schur functions, Stanley symmetric functions, and skew Schubert functions.
For general (not necessarily symmetric) edge-labeled posets P, it is natural to consider a quasi-symmetric generating function FP for maximal chains whose sequence of edge labels have fixed descents. We show this gives a Hopf morphism from an incidence algebra of edge-labeled posets to the Hopf algebra of quasi-symmetric functions. We also show this generating function is a generalization of a generating function for flag f-vectors defined by Ehrenborg and, for symmetric edge-labeled posets, it coincides with the symmetric function of the previous paragraph.