Thec-2d-Index of Oriented Matroids

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

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