A Survey of Eulerian Posets

Abstract

An Eulerian poset is a finite graded poset with o and i such that every interval of length at least one has the same number of elements of odd rank as of even rank. For instance, the face lattice of a convex polytope is Eulerian. We survey some numerical and polynomial invariants associated with an Eulerian poset P. The flag f-vector counts the number of chains of P whose elements have specified ranks. A convenient way to represent the flag f-vector is by a noncommutative polynomial Φp(c, d) called the cd-index of P. The problem of characterizing the flag f-vector of certain classes of Eulerian posets, notably those which are Cohen-Macaulay, is best approached in the context of the cd-index. For the special class of simplicial Eulerian posets (which include face lattices of simplicial polytopes and triangulations of spheres), much more can be said about the flag f-vector. A high point of this subject is the g-theorem, which characterizes the f-vectors of simplicial convex polytopes. In Section 4 we discuss the concept of the h-vector of a lower Eulerian poset and its connection with intersection homology theory. The notion of h-vector leads naturally to the theory of acceptable functions on a lower Eulerian poset and their connection with subdivisions and the Ehrhart polynomial.