Catalan pairs and Fishburn triples

TL;DR

This paper introduces new relational structures called Catalan pairs of type 2 and Fishburn triples, linking Catalan and Fishburn objects, and generalizes known statistical distributions from Catalan to Fishburn structures.

Contribution

It defines Fishburn triples as a unifying framework for Catalan and Fishburn objects, extending existing combinatorial structures and their statistical properties.

Findings

Fishburn triples generalize Catalan pairs.

Several Catalan statistics are extended to Fishburn objects.

New equidistribution results for Fishburn statistics are established.

Abstract

Disanto, Ferrari, Pinzani and Rinaldi have introduced the concept of 'Catalan pair', which is a pair of partial orders (S,R) satisfying certain axioms. They have shown that Catalan pairs provide a natural description of objects belonging to several classes enumerated by Catalan numbers. In this paper, we first introduce another axiomatic structure (T,R), which we call the 'Catalan pair of type 2', which describes certain Catalan objects that do not seem to have an easy interpretation in terms of the original Catalan pairs. We then introduce 'Fishburn triples', which are relational structures obtained as a direct common generalization of the two types of Catalan pairs. Fishburn triples encode, in a natural way, the structure of objects enumerated by the Fishburn numbers, such as interval orders or Fishburn matrices. This connection between Catalan objects and Fishburn objects allows us to associate known statistics on Catalan objects with analogous statistics of Fishburn objects. As our main result, we then show that several known equidistribution results on Catalan statistics can be generalized to analogous results for Fishburn statistics.