I'm concerned here with Catalan numbers.
There are many combinatorial interpretation of these numbers. Here I would focus on the interpretation around words build with 2 symbols, let say [ and ]. The Nth Catalan number for such pair of symbols gives the number of words of length 2N such that any sub words never contains more [ than ].
I'm interested what we can say if we take not only 2 symbols but 4 for example. Let say [, ] and (, ).
Then Nth "Generalized" Catalan number would be interpreted as the number of words of length 2N build with these 4 symbols such that any sub word is, let say "simple Catalan" for the pair [,], or "simple Catalan" for the pair (,) or both. In other words for any sub words, there is a pair of symbols which are "simple Catalan" for this sub word.
Hope I'm clear enough.