Coproducts and the cd-Index

The linear span of isomorphism classes of posets, P, has a Newtonian coalgebra structure. We observe that the ab-index is a Newtonian coalgebra map from the vector space P to the algebra of polynomials in the noncommutative variables a and b. This enables us to obtain explicit formulas showing how the cd-index of the face lattice of a convex polytope changes when taking the pyramid and the prism of the polytope and the corresponding operations on posets. As a corollary, we have new recursion formulas for the cd-index of the Boolean algebra and the cubical lattice. Moreover, these operations also have interpretations for certain classes of permutations, including simsun and signed simsun permutations. We prove an identity for the shelling components of the simplex. Lastly, we show how to compute the ab-index of the Cartesian product of two posets given the ab-indexes of each poset.