Thec-2d-Index of Oriented Matroids

We obtain an explicit method to compute the cd-index of the lattice of regions of an oriented matroid from the ab-index of the corresponding lattice of flats. Since the cd-index of the lattice of regions is a polynomial in the ring Z(c, 2d), we call it the c-2d-index. As an application we obtain a zonotopal analogue of a conjecture of Stanley: among all zonotopes the cubical lattice has the smallest c-2d-index coefficient-wise. We give a new combinatorial description for the c-2d-index of the cubical lattice and the cd-index of the Boolean algebra in terms of all the permutations in the symmetric group Sn . Finally, we show that only two-thirds of the :(S)'s of the lattice of flats are needed for the c-2d-index computation. 1997 Academic Press