THE C-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 p olynomial in the ring Z[c, 2d], we call it the c-2d-index. As an appli cation we obtain a zonotopal analogue of a conjecture of Stanley: amon g all zonotopes the cubical lattice has the smallest c-2d-index coeffi cient-wise. We give a new combinatorial description for the c-2d-index of the cubical lattice and the ed-index of the Boolean algebra in ter ms of all the permutations in the symmetric group S-n. Finally, we sho w that only two-thirds of the alpha(S)'s of the lattice of flats are n eeded for the c-7d-index computation. (C) 1997 Academic Press.