TITLE: Color an $n\times n$ square with $n$ colors
QUESTION [15 upvotes]: How many ways is there to color an $n\times n$ square grid with $n$ colors such that each column and each row contains exactly one $1\times 1$ square of each color?
And how many ways if the same is required of the two diagonals?

REPLY [6 votes]: In section 7 of this survey paper, Douglas Stones writes:

Comtet said that even estimating $L_n$ when $n → ∞$ “seems to be an
  extremely difficult combinatorial problem.” However, van Lint and Wilson showed that
  $$\frac{1}{n}
L_n^{1/n^2}
→ \exp(−2).$$ ... This is not a particularly satisfying result ... 
... there is still a long way to go to achieve a ‘clean’ asymptotic solution for the
  number of Latin squares.