Mathematical Properties and Interactions of Catalan Numbers and Symmetry Groups

This study focuses on the history of Catalan numbers, including definitions, formulas, combinatorial meanings, and geometric interpretations. Catalan numbers are a sequence of natural numbers widely used in combinatorial mathematics, often used to represent arrangements of structures such as balanced brackets, binary trees and polygonal triangular dissections. In terms of the definition and properties of symmetry groups, the study includes the structure of orders and subgroups as well as representation theory. Symmetry groups describe the symmetry of geometric objects, including rotations, reflections, and axes of rotational reflections. The connection between Catalan numbers and symmetry groups, including influences and interactions, is further analysed.