Extending Ziegler's Prism Recurrence Formula: A Closed-Form Expression for Face Enumeration of Prismatic Polytopes

We present a novel closed-form expression for the enumeration of k-dimensional faces in prismatically constructed polytopes formed via repeated Cartesian products with the unit interval. Building upon Ziegler’s classical recurrence relation for face counts under prism operations—namely, 𝑓𝑘(𝑃 × 𝐼) = 2 ⋅ 𝑓𝑘(𝑃) + 𝑓𝑘−1(𝑃) —we derive a binomial convolution formula that unrolls this recurrence over nnn dimensions. The resulting expression computes 𝑓𝑘 directly from the base polytope's face data and bypasses recursive computation. Our formula captures the combinatorial structure of the face lattice of these prismatic polytopes and aligns with the product rules outlined in Grünbaum’s Convex Polytopes. This work contributes a compact and computationally efficient method for face enumeration and offers insight into the algebraic structure underlying iterated polytope products.