Latin Rectangle
A 
Latin rectangle is a Matrix with elements 
such thatentries in each row and column are distinct. If 
, the special case of a Latin Square results. A normalizedLatin rectangle has first row 
and first column 
. Let 
be the numberof normalized Latin rectangles, then the total number of Latin rectangles is
(McKay and Rogoyski 1995), where
is a Factorial. Kerewala (1941) found a Recurrence Relation for
, and Athreya, Pranesachar, and Singhi (1980) found a summation Formula for 
.The asymptotic value of 
was found by Godsil and McKay (1990). The numbers of Latinrectangles are given in the following table from McKay and Rogoyski (1995). The entries 
and 
areomitted, since
 |  |  | |
 | |  |
but
and 
are included for clarity. The values of are given as a ``wrap-around'' seriesby Sloane's 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 |
References
Athreya, K. B.; Pranesachar, C. R.; and Singhi, N. M. ``On the Number of Latin Rectangles and Chromatic Polynomial of Colbourn, C. J. and Dinitz, J. H. (Eds.) CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, 1996. Godsil, C. D. and McKay, B. D. ``Asymptotic Enumeration of Latin Rectangles.'' J. Combin. Th. Ser. B 48, 19-44, 1990. Kerawla, S. M. ``The Enumeration of Latin Rectangle of Depth Three by Means of Difference Equation'' [sic]. Bull. Calcutta Math. Soc. 33, 119-127, 1941. McKay, B. D. and Rogoyski, E. ``Latin Squares of Order 10.'' Electronic J. Combinatorics 2, N3 1-4, 1995.http://www.combinatorics.org/Volume_2/volume2.html#N3. Ryser, H. J. ``Latin Rectangles.'' §3.3 in Combinatorial Mathematics. Buffalo, NY: Math. Assoc. of Amer., pp. 35-37, 1963. Sloane, N. J. A. Sequence A001009in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
.'' Europ. J. Combin. 1, 9-17, 1980.
- Orthogonal Rotation Group
- Orthogonal Tensors
- Orthogonal Transformation
- Orthogonal Vectors
- Orthographic Projection
- Orthologic
- Orthonormal Basis
- Orthonormal Functions
- Orthonormal Vectors
- Orthopole
- Orthoptic Curve
- Orthotomic
- Orthotope
- Osborne's Rule
- Oscillation
- Oscillation Land
- Osculating Circle
- Osculating Curves
- Osculating Interpolation
- Osculating Plane
- Osculating Sphere
- Osedelec Theorem
- Ostrowski's Inequality
- Ostrowski's Theorem
- Otter's Tree Enumeration Constants