Catalan structures and Catalan pairs

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 in [DFPR] 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. 1 Catalan pairs Catalan numbers are a very popular sequence of integer numbers, arising in many combinatorial problems coming out from different scientific areas, including computer science, computational biology, and mathematical physics [St1, St2]. Throughout all the paper, we will refer to any combinatorial structure enumerated by Catalan numbers as to a Catalan structure. Catalan pairs have been introduced in [DFPR] with the aim of giving a common language to (almost) all the Catalan structures. To reach this goal the authors of [DFPR] use an elementary mathematical tool, the Catalan pair, which is substantially a pair of binary relations satisfying certain axioms. They first prove that Catalan pairs are enumerated by Catalan numbers, according to their size. Their main goal is to prove that almost all Catalan structures can be interpreted in terms of Catalan pairs, thus providing a powerful tool to determine, in automatic way, many bijections between Catalan structures. Still in [DFPR], the authors prove several combinatorial properties of such Catalan pairs, also showing that they are related to some classes of pattern avoiding posets. In this paper we carry on the original purpose of [DFPR], and attempt at developing a general method to determine a representation of a given Catalan structure in terms of Catalan pairs. Our method relies on the observation that most of the Catalan structures admit a recursive decomposition, which can be naturally translated onto the two binary relations defining Catalan pairs. Once we have presented our methodology, in Section 2 we apply it to furnish the interpretation of some of the most classical Catalan structures in terms of Catalan pairs. In the final section, we extend our method in order to include some other Catalan structures. 1.1 Basic definitions In what follows we recall from [DFPR] some basic definitions of Catalan pairs. Given any set X, we denote D = D(X) the diagonal of X, that is the relation D = {(x, x) | x ∈ X}. Dipartimento di Sistemi e Informatica, viale Morgagni 65, 50134 Firenze, Italy [email protected] [email protected] Dipartimento di Scienze Matematiche ed Informatiche, Pian dei Mantellini, 44, 53100, Siena, Italy [email protected] [email protected]