## Latin Rectangle

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

(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 Latin rectangles are given in the following table from McKay and Rogoyski (1995). The entries and are omitted, since

but and are included for clarity. The values of are given as a wrap-around'' series by 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 .'' Europ. J. Combin. 1, 9-17, 1980.

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 A001009 in The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.