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!
Best Answer
When you count the number of positively directed paths from $(0,0,\dots,0)$ to $(n,n,\dots,n)$ that lie in the region $x_d\le x_1+\cdots+x_{d-1}$, you can project to the plane $(x_d,x_1+\cdots+x_{d-1})$ and find that you need the number of planar paths from $(0,0)$ to $(n,n(d-1))$ which stay above the line $x=y$, and which have $n$ vertical steps of each color {1,...,d-1}. So the answer comes to be $$\left(\binom{nd}{n}-\binom{nd}{n-1}\right)\binom{n(d-1)}{n,\dots,n}=\frac{n(d-2)+1}{n(d-1)+1}\frac{(nd)!}{(n!)^d}.$$ See also this previous question for enumerating lattice paths below a line.
On the other hand, one way to interpret Catalan numbers as lattice paths below the diagonal is to look at it as counting the number of standard Young tableaux of shape $(n,n)$. So a natural generalization is for example the number of standard Young tableaux of shape $(n,n,\dots,n)$. This corresponds to the region $x_1\le x_2\le\cdots\le x_d$, and can be counted using the hook-length formula. See my answer here.