Combinatorial properties of Catalan pairs

We define the notion of a Catalan pair, which is a pair of (strict) order relations (S, R) satisfying certain axioms. We show that Catalan pairs of size n are counted by Catalan numbers. We study some combinatorial properties of the relations R and S. In particular, we show that the second component R uniquely determines the pair, and we give a characterization of the poset R in terms of forbidden configurations. We also propose some generalizations of Catalan pairs arising from the modification of one of the axioms.