Inequalities for cd-Indices of Joins and Products of Polytopes

The cd-index is a polynomial which encodes the flag f-vector of a convex polytope. For polytopes U and V, we determine explicit recurrences for computing the cd-index of the free join $U\textcircled{v}V$ and the cd-index of the Cartesian product U x V. As an application of these recurrences, we prove the inequality $\psi {\left( {U{\kern 1pt} \textcircled{v}{\left( {V \times W} \right)}} \right)} \leqslant \psi {\left( {{\left( {U\textcircled{v}V} \right)} \times W} \right)}$ involving the cd-indices of three polytopes.