Quasisymmetric functions for nestohedra

Abstract

For a generalized permutohedron Q the enumerator F(Q) of positive lattice points in interiors of maximal cones of the normal fan ΣQ is a quasisymmetric function. We describe this function for the class of nestohedra as a Hopf algebra morphism from a combinatorial Hopf algebra of building sets. For the class of graph-associahedra, the corresponding quasisymmetric function is a new isomorphism invariant of graphs. The obtained invariant is quite natural as it is the generating function of ordered colorings of graphs and it satisfies the recurrence relation with respect to deletions of vertices.