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) \(\mathbf{c}\mathbf{d}\)-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 \(\mathbf{c}\mathbf{d}\)-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 \(\mathbf{c}\mathbf{d}\)-index for acyclic digraphs having a balanced linear edge labeling.

我们通过放宽分级，波塞和欧拉要求，将链枚举归纳为分级的部分有序集合。得到的平衡有向图，包括带有R标签的经典欧拉体，意味着存在（非同质）\（\ mathbf {c} \ mathbf {d} \）- index，这是研究不等式的关键不变式多面体的标志向量。镜像欧拉体形的亚历山大对偶，我们展示了有界平衡有向图的亚历山大对偶。对于重要的平衡图族Coxeter组的Bruhat图，我们的理论给出了完整\（\ mathbf {c} \ mathbf {d} \）存在的基本证明。-index及其属性。我们还介绍了带标记的无环有向图的上升和下降准对称函数，并表明它们是将平衡图向Stembridge峰代数映射的Hopf代数同态。我们猜想具有平衡的线性边缘标记的无环有向图的\（\ mathbf {c} \ mathbf {d} \）索引的非负性。