Recursive bijections for Catalan objects

TL;DR

This paper explores recursive and local bijections between different combinatorial structures counted by Catalan numbers, addressing questions posed by R. Stanley and providing insights into their relationships.

Contribution

It introduces new recursive and local bijections between Catalan-structured sets, enhancing understanding of their combinatorial correspondences.

Findings

Established recursive bijections between Catalan objects

Compared recursive and local bijection methods

Addressed open questions from R. Stanley's textbook

Abstract

In this note we introduce several instructive examples of bijections found between several different combinatorially defined sequences of sets. Each sequence has cardinalities given by the Catalan numbers. Our results answer some questions posed by R. Stanley in the addendum to his textbook. We actually discuss two types of bijection, one defined recursively and the other defined in a more local, relative, fashion. It is interesting to compare the results of the two.