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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.
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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.
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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.