I am studying Catalan numbers recently but I think that how about 3-D Catalan? So that I imagine following situation ;
A man travel through the path-way parallel to $ x, y, z $ axis from O $(0, 0, 0)$ to $P (n, n, n)$ $ (n \in N)$
However he chooses the Points $(x, y, z)$ satisfy $ x \leq y \leq z $ , and the path-way is always the shortest way $(x, y, z \in Z)$ during all his trip. What number of path-ways he can choose in his trip?
Abobe situation , This is my question.
The problem from 2017 Cayley Contest of CEMC of University of Waterloo is about 3D Catalan Number. The answer is: $\frac{2\cdot(3n)!}{n!\cdot(n+1)!\cdot(n+2)!} = \frac{2\cdot9!}{3!\cdot4!\cdot5!} = 42$