Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan Numbers

TL;DR

This paper introduces d-dimensional prime Catalan numbers, studies trapezoidal diagrams of crossing-free graphs, and establishes their relation to 3D prime Catalan numbers with exact growth rates.

Contribution

It defines prime Catalan numbers and trapezoidal diagrams, and links their enumeration to 3D prime Catalan numbers with precise exponential growth rates.

Findings

Exact growth rate for perfect matchings: 5.196^n

Exact growth rate for triangulations: 23.459^n

Lower bounds for embeddings of trapezoidal diagrams

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. We further introduce the notion of a trapezoidal diagram of a crossing-free geometric graph, such as a triangulation or a crossing-free perfect matching. 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 3 decimal places). Finally, we give lower bounds for the maximum number of embeddings of a trapezoidal diagram on any given point set.