Hopf algebra of building sets

TL;DR

This paper introduces a Hopf algebra structure on building sets, extending graph chromatic functions, and characterizes eulerian building sets with applications to symmetric functions and combinatorial invariants.

Contribution

It constructs a Hopf algebra on building sets, defines eulerian building sets, and characterizes them via intersection posets, extending graph theory concepts.

Findings

Construction of a Hopf algebra on building sets

Definition and characterization of eulerian building sets

Existence of the c d-index for eulerian building sets

Abstract

The combinatorial Hopf algebra on building sets $BSet$ extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the corresponding hypergraph. By passing from graphs to building sets, we construct a sequence of symmetric functions associated to a graph. From the generalized Dehn-Sommerville relations for the Hopf algebra $BSet$, we define a class of building sets called eulerian and show that eulerian building sets satisfy Bayer-Billera relations. We show the existence of the $\mathbf{c}\mathbf{d}-$index, the polynomial in two noncommutative variables associated to an eulerian building set. The complete characterization of eulerian building sets is given in terms of combinatorics of intersection posets of antichains of finite sets.