Let's call a nice matrix a square matrix of size $n$ with elements from $\{1,...,n\}$ such that every row and every column contains all the number $1,...,n$, 
What is the number of nice matrices ? 


a possible generalization is the following: the elements of the matrix are exactly $\{1,...,n^2\}$ and we ask that the sum of the element of the rows are equal. 


always the question what is their number? 


All comments are welcomed!
Thanks in advance



Number of Nice Matrices
Ok, the posibillity of no formula existing fascinated me so I followed Wolfgang's advise (even if it was not meant for me) and found a formula on the internet. I quote: 



> 
> Theorem 3. Let $p(z)$ be any monic polynomial of degree $n$ and let $M\_n$ be the family of
>  all $n × n$ matrices over $\{−1,+1\}$. Then 
>  $$L\_n = 2^{−n^2} \sum\_{X \in M\_n} p(\textrm{Per} X) \pi(X),$$
>  where Per$(X)$ is the permanent of $X$ and $\pi(X)$ is the product of the entries of $X$.
> 
> 
> 


From: On the number of Latin squares by Brendan D. McKay and Ian M. Wanless.


However the following quote from the same article is maybe more relevant in some respects:



> 
> The literature contains quite a few exact formulas for $R\_n$, but none of them appear very efficient for explicit computation (though Saxena [16] managed to compute $R\_7$ using such a formula).
> 
> 
> 


($L\_n$ is the number of nice matrices, $R\_n$ is the number of nice matrices whose first row and column read $1, \ldots, n$. $L\_n$ can easily be computed from $R\_n$ but is much bigger. The problem here more than anything else is that the numbers $R\_n$ and $L\_n$ grow perversely fast with $n$.)


