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 |