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Balanced and Bruhat graphs. (English) Zbl 07271138
Summary: We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an \(R\)-labeling, imply the existence of the (non-homogeneous) \( \mathbf{cd} \)-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for bounded balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existence of the complete \(\mathbf{cd} \)-index and its properties. We also introduce the rising and falling quasisymmetric functions of a labeled acyclic digraph and show they are Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak algebra. We conjecture non-negativity of the \(\mathbf{cd} \)-index for acyclic digraphs having a balanced linear edge labeling.
MSC:
06A11 Algebraic aspects of posets
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
05E05 Symmetric functions and generalizations
16T15 Coalgebras and comodules; corings
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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