Higher-dimensional Catalan numbers?

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.

The closest thing I can think of to what you want is triangulations of even dimensional cyclic polytopes. These are nice because, unlike triangulations of the cube, they all use the same number of simplices. See the above linked paper for more.

There are a number of other important generalizations of Catalan numbers, but these are the only ones I would particularly call higher-dimensional.

Two papers that both address your question along similar lines are:

(1) Free n-category generated by a cube, oriented matroids, and higher Bruhat orders by Karpranov and Voevodsky and

(2) Arrangements of hyperplanes, higher braid groups and higher Bruhat orders by Manin and Schechtman

Maps between higher Bruhat orders and higher Stasheff-Tamari posets by H. Thomas serves as a follow-up and fills in in some of the details, but has a different flavor.

Briefly, paper (1) defines (among other things) a poset $S(n,k), 0 \leq k \leq n$, with $|S(n,2)|$ equal to the $n^{th}$ Catalan number. The objects of this poset are defined in 3 ways (and shown to be equivalent): combinatorially (via cyclic polytopes), as "homotopies of homotopies" and as elements of "the free $(n-k)$-category on the $n$-simplex".

So if you like these definitions and think they are natural, $|S(n,k)|$ for $k >2$ gives you "higher dimensional" Catalan numbers.

Paper (2) does something nice, too, but I can't remember what.