University of Minnesota
Abstract: In this talk we will discuss various bases for
f-vectors of Eulerian posets. (Eulerian posets include, for example, face
lattices of convex polytopes and intervals in the strong Bruhat order).
In particular we will define the cd-index, a non-commutative polynomial
which compactly encodes the flag f-vector. The cd-index is conjectured
to have non-negative coefficients for any poset whose order complex is a
homology sphere. Using a series of basis changes, we can "lift" a trivial
convolution formula on flag f-vectors to a convolution formula on
cd-coefficients, leading to partial results on the non-negativity
conjecture. If time allows, we will also discuss the partial orders which
mediate many of the basis changes. These are natural partial orders whose
groundsets are counted by the Fibonacci numbers.