
Vladimir Grujić

Tanja Stojadinović
Keywords:
Hopf algebra, building set, graph, symmetric function, DehnSommerville relations, cdindex, simplicial complex
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 quasisymmetric 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 DehnSommerville relations for the Hopf algebra $BSet$, we define a class of building sets called eulerian and show that eulerian building sets satisfy BayerBillera 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.