Catalan structures and Catalan pairs

TL;DR

This paper introduces a method to translate recursive definitions of Catalan structures into Catalan pairs, providing a unified framework to interpret various Catalan-counted objects through binary relations.

Contribution

It presents an automatic approach to derive Catalan pairs from Catalan structures, linking recursive definitions to relation-based representations across multiple structures.

Findings

Unified interpretation of Catalan structures via binary relations

Application to well-known Catalan objects

Enhanced understanding of Catalan combinatorial objects

Abstract

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.