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.
@article{10_37236_2413,
author = {Vladimir Gruji\'c and Tanja Stojadinovi\'c},
title = {Hopf algebra of building sets.},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {4},
doi = {10.37236/2413},
zbl = {1279.16027},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2413/}
}
TY - JOUR
AU - Vladimir Grujić
AU - Tanja Stojadinović
TI - Hopf algebra of building sets.
JO - The electronic journal of combinatorics
PY - 2012
VL - 19
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/2413/
DO - 10.37236/2413
ID - 10_37236_2413
ER -
%0 Journal Article
%A Vladimir Grujić
%A Tanja Stojadinović
%T Hopf algebra of building sets.
%J The electronic journal of combinatorics
%D 2012
%V 19
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/2413/
%R 10.37236/2413
%F 10_37236_2413
Vladimir Grujić; Tanja Stojadinović. Hopf algebra of building sets.. The electronic journal of combinatorics, Tome 19 (2012) no. 4. doi: 10.37236/2413
