Open: the site labels the problem OPEN. Its commentary records the asymptotic $L_{k,n}\sim e^{-\binom k2}(n!)^k$ of Erdős and Kaplansky [ErKa46] for $k=o((\log n)^{3/2-\epsilon})$ and Yamamoto's extension of it to $k\le n^{1/3-o(1)}$ [Ya51], and calls sequence A001009 of the OEIS the count of such Latin rectangles. That sequence lists the normalized counts $R_{k,n}$, rectangles whose first row and first column are in natural order, with $L_{k,n}=\dfrac{n!\,(n-1)!}{(n-k)!}\,R_{k,n}$ for the labeled count $L_{k,n}$ that the asymptotics use.