let's try listing a bunch of situations where n! counts something and catalan(n) counts some "smaller" related thing...

n! counts permutations, while catalan(n) counts permutations which "don't zig-zag too much"... ??...

n! counts a-series schubert varieties, catalan(n) counts their tangent spaces at the basepoint... or something.

n! is the dimension of the a-series "hecke algebra", catalan(n) is the dimension of its image acting on... something... related to "quantum su(2)" ....