A Catalan question

This week I am nominally at the Permutation Patterns conference. I find myself on the edge of the (very strong) community of Permutation Patterners; also, after a month away, I have lots of catching up to do; also, Bruce Sagan is not here. So I am not the most regular attendee. Nevertheless, looking in from outside, I have formulated a question.

I explained in the preceding post how permutations can be represented by pairs of total orders. This gives a bijection between permutations and a class C of relational structures such that

• C is a Fraïssé class (so there is a unique countable homogeneous universal “permutation”, and we can ask questions about Ramsey properties etc.);
• the permutation patterners’ notion of subpattern translates exactly into the notion of induced substructure;
• the counting function for C is an interesting integer sequence (in this case, the factorials).

In the talks I have attended, the Catalan numbers have figured very prominently; they come in to counting meandric systems, interval posets, first-order properties of permutations, partial shuffles, Catalan words, and bumps, to name just some. A wordcloud for the conference talks would certainly feature the word “Catalan”.

So my question is as follows. Is there a class C of relational structures which are bijective with a class of objects of interest to permutation patterners such that

• C is a Fraïssé class;
• the permutation patterners’ notion of subobjects translates exactly into the notion of induced substructure;
• the counting function for C is the sequence of Catalan numbers?

The difficulty is that there are so many occurrences of the Catalan numbers that I cannot properly survey them all!