an:07271138
Zbl 7271138
Ehrenborg, Richard; Readdy, Margaret
Balanced and Bruhat graphs
EN
Ann. Comb. 24, No. 3, 587-617 (2020)
0218-0006 0219-3094
2020
j
06A11 52B05 05E05 06A08 16T15 20F55
Alexander duality; balanced digraph; Bruhat graph; \( \mathbf{cd} \)-index; Eulerian poset; quasisymmetric function; \(R\)-labeling
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.