Cyclic inclusion-exclusion

TL;DR

This paper introduces cyclic inclusion-exclusion, a combinatorial operation that characterizes the kernel of a morphism from acyclic graphs to quasi-symmetric functions, with applications to Kerov character polynomials.

Contribution

It defines cyclic inclusion-exclusion and describes its role in understanding the kernel of a graph-to-quasi-symmetric function morphism, extending to noncommutative and bipartite cases.

Findings

Characterizes the kernel of the morphism using cyclic inclusion-exclusion.

Extends the framework to noncommutative and bipartite graphs.

Provides an application to Kerov character polynomials.

Abstract

Following the lead of Stanley and Gessel, we consider a morphism which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph. We describe the kernel of this morphism, using a simple combinatorial operation that we call cyclic inclusion-exclusion. Our result also holds for the natural noncommutative analog and for the commutative and noncommutative restrictions to bipartite graphs. An application to the theory of Kerov character polynomials is given.