Catalan structures and Catalan pairs

Understand

A Catalan pair is a pair of binary relations (S,R) satisfying certain axioms.

• These objects are enumerated by the well-known Catalan numbers, and have been introduced with the aim of giving a common language to most of the structures counted by Catalan numbers.
• Here, we give a simple method to pass from the recursive definition of a generic Catalan structure to the recursive definition of the Catalan pair on the same structure, thus giving an automatic way to interpret Catalan structures in terms of Catalan pairs.
• We apply our method to many well-known Catalan structures, focusing on the meaning of the relations S and R in each considered case.