A 
Latin rectangle is a 
matrix with elements 
such that entries
in each row and column are distinct. If 
, the special case of a Latin
square results. A normalized Latin rectangle has first row 
and first column 
. Let 
be the number of normalized 
Latin rectangles, then the total number of 
Latin rectangles is
 |
(1)
|
(McKay and Rogoyski 1995), where 
is a factorial. Kerewala (1941)
found a recurrence relation for 
, and Athreya et al. (1980) found a summation formula for 
.
The asymptotic value of 
was found by Godsil and McKay (1990). The numbers
of 
Latin rectangles are given in the following table from McKay and Rogoyski (1995).
The entries 
and 
are omitted, since
 |  |  |
(2)
|
 |  |  |
(3)
|
but 
and 
are included for clarity. The values of 
are given as a "wrap-around" series by OEIS
A001009.
 |  |  |
| 1 | 1 | 1 |
| 2 | 1 | 1 |
| 3 | 2 | 1 |
| 4 | 2 | 3 |
| 4 | 3 | 4 |
| 5 | 2 | 11 |
| 5 | 3 | 46 |
| 5 | 4 | 56 |
| 6 | 2 | 53 |
| 6 | 3 | 1064 |
| 6 | 4 | 6552 |
| 6 | 5 | 9408 |
| 7 | 2 | 309 |
| 7 | 3 | 35792 |
| 7 | 4 | 1293216 |
| 7 | 5 | 11270400 |
| 7 | 6 | 16942080 |
| 8 | 2 | 2119 |
| 8 | 3 | 1673792 |
| 8 | 4 | 420909504 |
| 8 | 5 | 27206658048 |
| 8 | 6 | 335390189568 |
| 8 | 7 | 535281401856 |
| 9 | 2 | 16687 |
| 9 | 3 | 103443808 |
| 9 | 4 | 207624560256 |
| 9 | 5 | 112681643083776 |
| 9 | 6 | 12952605404381184 |
| 9 | 7 | 224382967916691456 |
| 9 | 8 | 377597570964258816 |
| 10 | 2 | 148329 |
| 10 | 3 | 8154999232 |
| 10 | 4 | 147174521059584 |
| 10 | 5 | 746988383076286464 |
| 10 | 6 | 870735405591003709440 |
| 10 | 7 | 177144296983054185922560 |
| 10 | 8 | 4292039421591854273003520 |
| 10 | 9 | 7580721483160132811489280 |
." Europ. J. Combin. 1,
9-17, 1980.Colbourn, C. J. and Dinitz, J. H. (Eds.).