Extending Ziegler's Prism Recurrence Formula: A Closed-Form Expression for Face Enumeration of Prismatic Polytopes
Year: 2025
Type: article
Abstract: 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 Zieglerโs classical recurrence relation for face counts under prism operationsโnamely, ๐๐(๐ ร ๐ผ) = 2 โ
๐๐(๐) + ๐๐โ1(๐) โwe derive a binomial convolution formula that unrolls this recurrence over nnn dimensions. The resulting expression computes ๐๐ 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 Grรผnbaumโs 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.
Author: Narasimha, Jayanth
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7082060031
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