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 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 Σ k on graded posets and show that relations between the ab -indices (or cd -indices, for an Eulerian poset P ) of barycentric subdivisions of Σ 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 of regions of the arrangement in terms of the cd -indices of and .