Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan Numbers

Abstract

The d-dimensional Catalan numbers form a well-known sequence of numbers which count balanced bracket expressions over an alphabet of size d. In this paper, we introduce and study what we call d-dimensional prime Catalan numbers, a sequence of numbers which count only a very specific subset of indecomposable balanced bracket expressions. These numbers were encountered during the investigation of what we call trapezoidal diagrams of geometric graphs, such as triangulations or crossing-free perfect matchings. In essence, such a diagram is obtained by augmenting the geometric graph in question with its trapezoidal decomposition, and then forgetting about the precise coordinates of individual vertices while preserving the vertical visibility relations between vertices and segments. We note that trapezoidal diagrams of triangulations are closely related to abstract upward triangulations. We study the numbers of such diagrams in the cases of (i) perfect matchings and (ii) triangulations. We give bijective proofs which establish relations with 3-dimensional (prime) Catalan numbers. This allows us to determine the corresponding exponential growth rates exactly as (i) $$5.196^n$$ and (ii) $$23.459^n$$ (bases are rounded to three decimal places). Finally, we give exponential lower bounds for the maximum number of embeddings that a trapezoidal diagram can have on any given point set.