Catalan numbers and relations

TL;DR

This paper introduces Catalan pairs, a unified framework using binary relations to interpret Catalan numbers combinatorially, and explores their properties and generalizations.

Contribution

It defines Catalan pairs with axioms, characterizes the second relation, and proposes generalizations, providing a unified language for Catalan number interpretations.

Findings

The second relation R uniquely determines the pair.

Characterization of R via forbidden configurations.

Proposed generalizations of Catalan pairs.

Abstract

We define the notion of a Catalan pair (which is a pair of binary relations (S,R) satisfying certain axioms) with the aim of giving a common language to most of the combinatorial interpretations of Catalan numbers. We show, in particular, that the second component R uniquely determines the pair, and we give a characterization of R in terms of forbidden configurations. We also propose some generalizations of Catalan pairs arising from some slight modifications of (some of the) axioms.