We prove that the cd-index of a convex polytope satisfies a strong monotonicity property with respect to the cd-indices of any face and its link. As a consequence, we prove for d-dimensional polytopes a conjecture of Stanley that the cd-index is minimized on the d-dimensional simplex. Moreover, we prove the upper bound theorem for the cd-index, namely that the cd-index of any d-dimensional poly...

A graded partially ordered set is Eulerian if every interval has the same number of elements of even rank and of odd rank. Face lattices of convex polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector can be encoded efficiently in the cd-index. The cd-index of a polytope has all positive entries. An important open problem is to give the broadest natural class of Eulerian ...

where hk = dim(A(∆)⊗AR) k. When the fan ∆ is rational, it corresponds to a toric variety X(∆) and the numbers hk are the even betti numbers of the cohomology of X(∆). In any case, they satisfy the Poincaré duality hk = hn−k. When the fan ∆ is complete but not necessarily simplicial, one can apply the same construction to the first barycentric subdivision B(∆) of ∆. We can label a 1-dimensional ...

We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals, using basic geometric operations which preserve PL sphericity and have a simple effect on the cd-index. This leads to a new proof that Bruhat intervals are P...

Let [u, v] be a Bruhat interval of a Coxeter group such that the Bruhat graph BG(u, v) of [u, v] is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of [u, v]. Since in this case the complete cd-index and the cd-index of [u, v] coincide, we also obtain a new combinatorial interpretation for the coefficients of t...

We investigate a special class of polytopes, the zonotopes, and show that their flag f -vectors satisfy only the affine relations fulfilled by flag f -vectors of all polytopes. In addition, we determine the lattice spanned by flag f -vectors of zonotopes. By duality, these results apply as well to the flag f -vectors of central arrangements of hyperplanes.

The number of flags in a complete fan, or more generally in an Eulerian poset, is encoded in the cd-index. We prove the non-negativity of the cd-index for complete fans, regular CW -spheres and Gorenstein* posets.

The cd-index is a polynomial which encodes the flag f-vector of a convex polytope. For poly-topes U and V , we determine explicit recurrences for computing the cd-index of the free join U ∨ V and the cd-index of the Cartesian product U × V. As an application of these recurrences, we prove the inequality Ψ(U ∨ (V × W)) ≤ Ψ((U ∨ V) × W) involving the cd-indices of three polytopes.

The cd-index is a polynomial which encodes the ag f-vector of a convex polytope. We determine explicit recurrences for computing the cd-index of the free join U _ V of two polytopes U and V and the cd-index of the Cartesian product U V. As an application of these recurrences, we prove the inequality (U _ (V W)) ((U _ V) W) involving the cd-indices of three polytopes.