Weighted derivations and the cd-index
Weighted derivations W(1) and W(2) allowed R. Ehrenborg and M. Readdy ( Discrete Comput. Geom. 21: 389-403, 1999) to give a recursive description of the cd-indices of the lattices of the regions of the arrangements A(n) and B(n). In part motivated by this, we describe a new basis for the subspace spanned by ab-indices of all simplicial graded posets and determine the action of certain linear maps (associated with weighted derivations W(k)) on this basis. Extending the "pyramid" and "prism" operations, we define operations Sigma(k) on graded posets and show that relations between the ab-indices ( or cd-indices, for an Eulerian poset P) of barycentric subdivisions of Sigma(k)(P) and P can be described by using the linear maps associated by weighted derivations W(k). Finally, in response to a question from Ehrenborg and Readdy (Discrete Comput. Geom. 21: 389-403, 1999), we obtain the formulae that express the cd-index of the lattice R(Dn) of regions of the arrangement D(n) in terms of the cd-indices of R(Bn) and R(An-2).
cd-index eulerian posets barycentric subdivisions ab-index simplicial poset h-vector