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