I am trying to figure out if the following expression

$$\frac{(n^2 - n)! }{ n! ((n-1)!)^n }$$

is an integer for all positive integer $n.$

I tried the induction, but induction case is running into problem. So I was looking at it from permutation/combination problems. But so far, couldn't come up with a convincing argument.

Does anybody has any thoughts on this to share ?

Thank you very much.