Akademik Veri Yönetim Sistemi
MATHEMATICS, cilt.13, sa.23, ss.1-18, 2025 (SCI-Expanded)
Polygon triangulations and their generalizations to angulations
are fundamental in combinatorics and computational geometry. This paper
presents a unified linear-time framework that establishes explicit
bijections between Dyck words, planted ary trees, and angulations
of convex polygons. We introduce stack-based and tree-based algorithms
that enable reversible conversion between symbolic and geometric
representations, prove their correctness and optimal complexity, and
demonstrate their scalability through extensive experiments. The
approach reveals a hierarchical decomposition encoded by Fuss–Catalan
numbers, providing a compact and uniform representation for
triangulations, quadrangulations, pentangulations, and higher-arity
angulations. Experimental comparisons show clear advantages over
rotation-based methods in both runtime and memory usage. The framework
offers a general combinatorial foundation that supports efficient
enumeration, compressed representation, and extensions to
higher-dimensional or non-convex settings.