TITLE: Higher-dimensional Catalan numbers?
QUESTION [24 upvotes]: One could imagine defining various notions of higher-dimensional Catalan numbers,
by generalizing objects they count.
For example, because the Catalan numbers count the triangulations of convex polygons,
one could count the tetrahedralizations of convex polyhedra, or more generally, triangulations of
polytopes.  Perhaps the number of triangulations of the $n$-cube are similar to the Catalan
numbers?
Or, because the Catalan numbers count the number of below-diagonal monotonic paths
in an $n \times n$ grid, one could count the number of monotonic paths below
the "diagonal hyperplane" in an $n \times \cdots \times n$ grid.
I would be interested in references to such generalizations, which surely have been considered—I am just not looking under the right terminology.
I would be particularly grateful to learn of generalizations that share some of the
Catalan number's ubiquity.  Thanks for pointers!

REPLY [2 votes]: One relatively direct generalization is to interpret the $n$-th Catalan number as the multiplicity of the trivial representation in the $2n$-th tensor power of the standard $2$-dimensional representation of $SU_2(\mathbf{C})$, and to consider instead the multiplicity of the trivial representation in the $kn$-th tensor power of the standard $k$-th dimensional representation of $SU_k(\mathbf{C})$.
(On the other hand, I don't known which other properties of the Catalan numbers will have any analogue here...)