What does the cd-index count?

The celebrated g-theorem characterizes the f-vectors of simplicial spheres. For more general cell decompositions of spheres — and, more generally, for Eulerian posets — the natural refinement is to count chains of faces: the flag f-vector. Its $2^d$ entries satisfy the generalized Dehn–Sommerville relations, and the cd-index compresses them, with no linear redundancy, into a Fibonacci number of coefficients. Stanley conjectured, and Karu proved via sheaf-cohomological methods, that these coefficients are non-negative for every Gorenstein* poset. A basic question remains open: what do the coefficients count? In this talk I will introduce S-partitionable posets, a class of Eulerian posets generalizing both Stanley's S-shellable spheres and partitionable simplicial complexes. For these posets the cd-index is non-negative, can be computed recursively, and counts the blocks of a partition of the faces of the order complex. I will also discuss a semi-Eulerian extension and several open problems. This is joint work with Dan Guyer (UW) and José Samper (PUC).