Generalized Dehn–Sommerville relations for hypergraphs

Abstract

Mathematics Subject Classifications: 16T05, 16T30, 05C65, 05E45 AbstractThe generalized Dehn-Sommerville relations determine the odd subalgebra of the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs that satisfy the generalized Dehn-Sommerville relations for the combinatorial Hopf algebra of hypergraphs. We characterize a wide class of eulerian hypergraphs according to the combinatorics of underlying clutters. The analogous results hold for simplicial complexes by the isomorphism which is induced from the correspondence of clutters and simplicial complexes.

Quasisymmetric Functions for Nestohedra

“…Find a graph or a building set that satisfies the generalized Dehn-Sommerville relations for B. The same problem is resolved in [10] for the chromatic Hopf algebra of hypergraphs, where the whole class of solutions called eulerian hypergraphs are found.…”

“…Remark 4.2. Another combinatorial Hopf algebra of building set BSet, which is a Hopf subalgebra of the chromatic Hopf algebra of hypergraphs is studied in [9], [10]. As algebras B and BSet are the same but the coalgebra structures are different.…”

Integer points enumerator of hypergraphic polytopes

“…A combinatorial Hopf algebra is a pair (H, ζ) of a graded connected Hopf algebra H = ⊕ n 0 H n over a field k, whose homogeneous components H n , n 0 are finitedimensional, and a multiplicative linear functional ζ : H → k called character. We consider a combinatorial Hopf algebra structure on hypergraphs different from the chromatic Hopf algebra of hypergraphs studied in [10]. The difference is in the coalgebra structures based on different combinatorial constructions, which is manifested in (non)co-commutativity.…”

“…Example 4.4. Simplicial complexes generate another Hopf subalgebra of HG which is isomorphic to the Hopf algebra of simplicial complexes introduced in [10] and studied more extensively in [3]. It is shown in [1,Lemma 21.2] that hypergraphic polytopes P K and P K 1 corresponding to a simplicial complex K and its 1-skeleton K 1 are normally equivalent and therefore have the same enumerators F q (P K ) = F q (P K 1 ).…”

“…The theory of combinatorial Hopf algebras developed by Aguiar, Bergeron and Sottile in the seminal paper [2] provides an algebraic framework for symmetric and quasisymmetric generating functions arising in enumerative combinatorics. Extensive studies of various combinatorial Hopf algebras are initiated recently [3,4,9,10]. The geometric interpretation of the corresponding (quasi)symmetric functions was first given for matroids [4] and then for simple graphs [6] and building sets [7].…”