On flag f-vectors of Gorenstein* posets

The flag f-vector is a basic invariant of finite graded posets, counting chains according to the set of ranks they occupy. For Eulerian posets it is efficiently encoded by the cd-index, a noncommutative polynomial in variables $c$ and $d$ with integer coefficients. Moreover, if the poset is Gorenstien*, in particular if the order complex of its proper part is topologically a sphere, then Karu proved Stanley's conjecture that all the coefficients of the cd-index are nonnegative. What more can be said about the cd-index of Gorenstein* posets? Upper bounds? A characterization?

We characterize the cd-index for rank 5 (lower ranks are easy, rank 5 corresponds to 3-dimensional spheres), obtain upper bounds for certain coefficients in all ranks, and conjecture further upper bounds for the entire cd-index.

Reflects joint with Satoshi Murai.

All needed background will be given in the talk!