OPEN

Give an asymptotic formula for the number of $k\times n$

Latin rectangles.

Erdős and Kaplansky

[ErKa46] proved the count is
\[\sim e^{-\binom{k}{2}}(n!)^k\]
when $k=o((\log n)^{3/2-\epsilon})$. Yamamoto

[Ya51] extended this to $k\leq n^{1/3-o(1)}$.

The count of such Latin rectangles is A001009 in the OEIS.