An Latin square is a Latin Rectangle with . Specifically, a Latin square consists of sets of the numbers 1 to arranged in such a way that no orthogonal (row or column) contains the same two numbers. The numbers of Latin squares of order , 2, ... are 1, 2, 12, 576, ... (Sloane's A002860). A pair of Latin squares is said to be orthogonal if the pairs formed by juxtaposing the two arrays are all distinct.

Two of the Latin squares of order 3 are

which are orthogonal. Two of the 576 Latin squares of order 4 are

A normalized, or reduced, Latin square is a Latin square with the first row and column given by
.
General Formulas for the number of *normalized* Latin squares are given by Nechvatal (1981),
Gessel (1987), and Shao and Wei (1992). The total number of Latin squares of order can then be computed from

The numbers of normalized Latin square of order , 2, ..., are 1, 1, 1, 4, 56, 9408, ... (Sloane's A000315). McKay and Rogoyski (1995) give the number of normalized Latin Rectangles for , ..., 10, as well as estimates for with , 12, ..., 15.

11 | |

12 | |

13 | |

14 | |

15 |

**References**

Colbourn, C. J. and Dinitz, J. H. *CRC Handbook of Combinatorial Designs.* Boca Raton, FL: CRC Press, 1996.

Gessel, I. ``Counting Latin Rectangles.'' *Bull. Amer. Math. Soc.* **16**, 79-83, 1987.

Hunter, J. A. H. and Madachy, J. S. *Mathematical Diversions.* New York: Dover, pp. 33-34, 1975.

Kraitchik, M. ``Latin Squares.'' §7.11 in *Mathematical Recreations.* New York: W. W. Norton, p. 178, 1942.

Lindner, C. C. and Rodger, C. A. *Design Theory.* Boca Raton, FL: CRC Press, 1997.

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.

Nechvatal, J. R. ``Asymptotic Enumeration of Generalised Latin Rectangles.'' *Util. Math.* **20**, 273-292, 1981.

Ryser, H. J. ``Latin Rectangles.'' §3.3 in *Combinatorial Mathematics.*
Buffalo, NY: Math. Assoc. Amer., pp. 35-37, 1963.

Shao, J.-Y. and Wei, W.-D. ``A Formula for the Number of Latin Squares.'' *Disc. Math.* **110**, 293-296, 1992.

Sloane, N. J. A. Sequences
A002860/M2051
and A000315/M3690
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-26