A 
Latin rectangle is a 
matrix over 
symbols, where 
are integers, with
entries 
distinct in each row and column. 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. 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 
.
Godsil and McKay (1990) obtained an asymptotic formula when 
.
The numbers of normalized 
Latin rectangles are given in the following table from
McKay and Rogoyski (1995). The entries 
and 
are omitted, since
 |  |  |
(2)
|
 |  |  |
(3)
|
where the second identity holds for 
. The values 
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.).