Thec-2d-Index of Oriented Matroids

Abstract

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 S n . 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

Manifold arrangements

“…In the manifold setting the computation of the cd-index now depends upon the intersection poset and the Euler characteristic of the elements of this quasi-graded poset. This extends the earlier studied spherical and toric arrangements [4,15].…”

Algebraic and Geometric Methods in Enumerative Combinatorics

“…Billera, Ehrenborg, and Readdy [BER97] extended this result to orientable matroids. They also asked for an interesting interpretation of ω(a Φ L (a, b)) for an arbitrary geometric lattice L; this is still open.…”

On Flag Vectors, the Dowling Lattice, and Braid Arrangements

“…Bayer and Sturmfels [5,Theorem 3.4] showed that the flag f -vector of the lattice of regions R depends only on the intersection lattice L. This dependency was showed in an explicit manner in [7].…”

“…Theorem 3.4 shows how to compute the c-2d-index of the lattice of regions. Theorem 3.4 [7]. Let H be a hyperplane arrangement, let R be the lattice of regions of H, and let L be the intersection lattice of H. Then the c-2d-index of R is given by…”

“…Then the characteristic polynomial is given by χ(P) = (q − 1) · E( (P)); see Proposition 5.3 of [7]. Apply the linear map E to Theorem 4.9.…”