In this paper we study finite Eulerian posets which are binomial, Sheffer or triangular. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows:
We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets;
We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases;
In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets.
We also study Eulerian triangular posets. This paper answers questions asked by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the boolean lattice by looking at smaller intervals.
Eulerian quasisymmetric functions and poset topology
We introduce a family of quasisymmetric functions called { \em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializ…
Alex Postnikov给出了Grassmannian的完全非负部分的组合显式胞腔剖分，记为Gr_{kn} +，并证明了这组胞腔作为分次偏序集与许多其他有趣的分次偏序集同构。本文的主要结果是一个显式的生成函数，它根据其维数枚举Gr_{kn} +中的细胞。作为推论，我们给出了Gr_{kn} +的欧拉示性数为1的新证明。此外，我们使用我们的结果来产生一个新的q模拟的欧拉数，插值之间的欧拉数，Narayana数，和二项式系数。
We introduce a family of quasisymmetric functions called { \em Eulerian quasisymmetric functions}, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surpris…
