Geodesic
Combinatorial Hopf Algebras of Simplicial Complexes
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

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We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley’s $(-1)$-color theorem. 
@article{DMTCS_2015_special_285_a50,
     author = {Benedetti, Carolina and Hallam, Joshua and Machacek, John},
     title = {Combinatorial {Hopf} {Algebras} of {Simplicial} {Complexes}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
     year = {2015},
     doi = {10.46298/dmtcs.2506},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2506/}
}
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Benedetti, Carolina; Hallam, Joshua; Machacek, John. Combinatorial Hopf Algebras of Simplicial Complexes. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi : 10.46298/dmtcs.2506. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2506/

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