I also found this interesting from a parser perspective:

https://en.wikipedia.org/wiki/Catalan_number

"Re-interpreting the symbol X as an open parenthesis and Y as a close
parenthesis, Cn counts the number of expressions containing n pairs of
parentheses which are correctly matched:
((()))     (()())     (())()     ()(())     ()()()
Cn is the number of different ways n + 1 factors can be completely parenthesized
(or the number of ways of associating n applications of a binary operator, as in
the matrix chain multiplication problem). For n = 3, for example, we have the
following five different parenthesizations of four factors:
((ab)c)d     (a(bc))d     (ab)(cd)     a((bc)d)     a(b(cd))"

(plus lots more interesting stuff in that entry)

- BE

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