Quasisymmetric functions for nestohedra

TL;DR

This paper introduces a new quasisymmetric function invariant for nestohedra and graph-associahedra, linking combinatorial geometry with algebraic structures and graph theory.

Contribution

It defines a Hopf algebra morphism for nestohedra and establishes a new graph invariant based on quasisymmetric functions.

Findings

The quasisymmetric function serves as a new isomorphism invariant for graphs.

The invariant is the generating function of ordered graph colorings.

It satisfies a recurrence relation with vertex deletions.

Abstract

For a generalized permutohedron $Q$ the enumerator $F(Q)$ of positive lattice points in interiors of maximal cones of the normal fan $\Sigma_Q$ 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 satisfies the recurrence relation with respect to deletions of vertices.