  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.
