Latin square

In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Here is an example:
A B C
C A B
B C A

The name "Latin square" is motivated by mathematical papers by Leonhard Euler, who used Latin characters as symbols. Of course, other symbols can be used instead of Latin letters: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3.

Reduced form

A Latin square is said to be reduced (also, normalized or in standard form) if both its first row and its first column are in their natural order. For example, the above Latin square is not reduced because its first column is A, C, B rather than A, B, C.

We can make any Latin square reduced by permuting (reordering) the rows and columns. Here switching the above matrix's second and third rows yields
A B C
B C A
C A B

which is reduced: Both its first row and its first column are alphabetically ordered A, B, C.
Properties
Orthogonal array representation

If each entry of an n × n Latin square is written as a triple (r,c,s), where r is the row, c is the column, and s is the symbol, we obtain a set of n² triples called the orthogonal array representation of the square. For example, the orthogonal array representation of the first Latin square displayed above is:

{ (1,1,1),(1,2,2),(1,3,3),(2,1,2),(2,2,3),(2,3,1),(3,1,3),(3,2,1),(3,3,2) },

where for example the triple (2,3,1) means that in row 2 and column 3 there is the symbol 1. The definition of a Latin square can be written in terms of orthogonal arrays:

A Latin square is the set of all triples (r,c,s), where 1 ≤ r, c, s ≤ n, such that all ordered pairs (r,c) are distinct, all ordered pairs (r,s) are distinct, and all ordered pairs (c,s) are distinct.

For any Latin square, there are n² triples since choosing any two uniquely determines the third. (Otherwise, an ordered pair would appear more than once in the Latin square.)

The orthogonal array representation shows that rows, columns and symbols play rather similar roles, as will be made clear below.
Equivalence classes of Latin squares

Many operations on a Latin square produce another Latin square (for example, turning it upside down).

If we permute the rows, permute the columns, and permute the names of the symbols of a Latin square, we obtain a new Latin square said to be isotopic to the first. Isotopism is an equivalence relation, so the set of all Latin squares is divided into subsets, called isotopy classes, such that two squares in the same class are isotopic and two squares in different classes are not isotopic.

Another type of operation is easiest to explain using the orthogonal array representation of the Latin square. If we systematically and consistently reorder the three items in each triple, another orthogonal array (and, thus, another Latin square) is obtained. For example, we can replace each triple (r,c,s) by (c,r,s) which corresponds to transposing the square (reflecting about its main diagonal), or we could replace each triple (r,c,s) by (c,s,r), which is a more complicated operation. Altogether there are 6 possibilities including "do nothing", giving us 6 Latin squares called the conjugates (also parastrophes) of the original square.

Finally, we can combine these two equivalence operations: two Latin squares are said to be paratopic, also main class isotopic, if one of them is isotopic to a conjugate of the other. This is again an equivalence relation, with the equivalence classes called main classes, species, or paratopy classes. Each main class contains up to 6 isotopy classes.
Number

There is no known easily-computable formula for the number L(n) of n × n Latin squares with symbols 1,2,...,n. The most accurate upper and lower bounds known for large n are far apart. One classic result is

\( \prod_{k=1}^n \left(k!\right)^{n/k}\geq L(n)\geq\frac{\left(n!\right)^{2n}}{n^{n^2}} \)

(this given by van Lint and Wilson).

Here we will give all the known exact values. It can be seen that the numbers grow exceedingly quickly. For each n, the number of Latin squares altogether (sequence A002860 in OEIS) is n! (n-1)! times the number of reduced Latin squares (sequence A000315 in OEIS).
The numbers of Latin squares of various sizes n reduced Latin squares of size n all Latin squares of size n

The numbers of Latin squares of various sizes

n	reduced Latin squares of size n	all Latin squares of size n
1	1	1
2	1	2
3	1	12
4	4	576
5	56	161280
6	9408	812851200
7	16942080	61479419904000
8	535281401856	108776032459082956800
9	377597570964258816	5524751496156892842531225600
10	7580721483160132811489280	9982437658213039871725064756920320000
11	5363937773277371298119673540771840	776966836171770144107444346734230682311065600000

For each n, each isotopy class (sequence A040082 in OEIS) contains up to (n!)³ Latin squares (the exact number varies), while each main class (sequence A003090 in OEIS) contains either 1, 2, 3 or 6 isotopy classes.
Equivalence classes of Latin squares n main classes isotopy classes

Equivalence classes of Latin squares

n	main classes	isotopy classes
1	1	1
2	1	1
3	1	1
4	2	2
5	2	2
6	12	22
7	147	564
8	283657	1676267
9	19270853541	115618721533
10	34817397894749939	208904371354363006
11	2036029552582883134196099	12216177315369229261482540

Examples

We give one example of a Latin square from each main class up to order 5.
\( \begin{bmatrix} 1 \end{bmatrix} \quad \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \quad \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{bmatrix}
\begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \\ 3 & 4 & 1 & 2 \\ 4 & 3 & 2 & 1 \end{bmatrix} \quad \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 1 & 3 \\ 3 & 1 & 4 & 2 \\ 4 & 3 & 2 & 1 \end{bmatrix}
\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 5 & 1 & 4 \\ 3 & 5 & 4 & 2 & 1 \\ 4 & 1 & 2 & 5 & 3 \\ 5 & 4 & 1 & 3 & 2 \end{bmatrix} \quad \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 1 & 5 & 3 \\ 3 & 5 & 4 & 2 & 1 \\ 4 & 1 & 5 & 3 & 2 \\ 5 & 3 & 2 & 1 & 4 \end{bmatrix} \)

They present, respectively, the multiplication tables of the following groups:

\( {0} \) – the trivial 1-element group
\( \mathbb{Z}_2 \) – the binary group
\( \mathbb{Z}_3 \) – cyclic group of order 3
\( \mathbb{Z}_2 \times \mathbb{Z}_2 \) – the Klein four-group
\( \mathbb{Z}_4 \) – cyclic group of order 4
\( \mathbb{Z}_5 \) – cyclic group of order 5
the last one is an example of a quasigroup, or rather a loop, which is not associative.

Applications

Latin squares are used in statistics and in mathematics.

In the design of experiments, Latin squares are a special case of row-column designs for two blocking factors:[1] Many row-column designs are constructed by concatenating Latin squares.[2]
In algebra, Latin squares are generalizations of groups; in fact, Latin squares are characterized as being the multiplication tables (Cayley tables) of quasigroups.

Error correcting codes

Sets of Latin squares that are orthogonal to each other have found an application as error correcting codes in situations where communication is disturbed by more types of noise than simple white noise, such as when attempting to transmit broadband Internet over powerlines.[3][4][5]

Firstly, the message is sent by using several frequencies, or channels, a common method that makes the signal less vulnerable to noise at any one specific frequency. A letter in the message to be sent is encoded by sending a series of signals at different frequencies at successive time intervals. In the example below, the letters A to L are encoded by sending signals at four different frequencies, in four time slots. The letter C, for instance, is encoded by first sending at frequency 3, then 4, 1 and 2.
\( \begin{matrix} A\\ B\\ C\\ D\\ \end{matrix} \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \\ 3 & 4 & 1 & 2 \\ 4 & 3 & 2 & 1 \\ \end{bmatrix} \quad \begin{matrix} E\\ F\\ G\\ H\\ \end{matrix} \begin{bmatrix} 1 & 3 & 4 & 2\\ 2 & 4 & 3 & 1\\ 3 & 1 & 2 & 4\\ 4 & 2 & 1 & 3\\ \end{bmatrix} \quad \begin{matrix} I\\ J\\ K\\ L\\ \end{matrix} \begin{bmatrix} 1 & 4 & 2 & 3\\ 2 & 3 & 1 & 4\\ 3 & 2 & 4 & 1\\ 4 & 1 & 3 & 2\\ \end{bmatrix} \)

The encoding of the twelve letters are formed from three Latin squares that are orthogonal to each other. Now imagine that there's added noise in channels 1 and 2 during the whole transmission. The letter A would then be picked up as:

\( \begin{matrix} 12 & 12 & 123 & 124\\ \end{matrix} \)

In other words, in the first slot we receive signals from both frequency 1 and frequency 2; while the third slot has signals from frequencies 1, 2 and 3. Because of the noise, we can no longer tell if the first two slots were 1,1 or 1,2 or 2,1 or 2,2. But the 1,2 case is the only one that yields a sequence matching a letter in the above table, the letter A. Similarly, we may imagine a burst of static over all frequencies in the third slot:

\( \begin{matrix} 1 & 2 & 1234 & 4\\ \end{matrix} \)

Again, we are able to infer from the table of encodings that it must have been the letter A being transmitted. The number of errors this code can spot is one less than the number of time slots. It has also been proved that if the number of frequencies is a prime or a power of a prime, the orthogonal Latin squares produce error detecting codes that are as efficient as possible.
Mathematical puzzles

The problem of determining if a partially filled square can be completed to form a Latin square is NP-complete.[6]

The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that nine particular 3×3 adjacent subsquares must also contain the digits 1–9 (in the standard version). The more recent KenKen puzzles are also examples of Latin squares.
Heraldry

The Latin square also figures in the arms of the Statistical Society of Canada,[7] being specifically mentioned in its blazon. Also, it appears in the logo of the International Biometric Society.[8]
See also

Latin hypercube sampling
Graeco-Latin square
Magic square
Problems in Latin squares
Small Latin squares and quasigroups
Mathematics of Sudoku
Futoshiki
Rook's graph, a graph that has Latin squares as its colorings
Eight queens puzzle
Block design
Word square

Notes

^
Bailey, R.A. (2008). "6 Row-Column designs and 9 More about Latin squares". Design of Comparative Experiments. Cambridge University Press. ISBN 978-0-521-68357-9. MR2422352. Pre-publication chapters are available on-line.
Hinkelmann, Klaus and Kempthorne, Oscar (2008). Design and Analysis of Experiments. I and II (Second ed.). Wiley. ISBN 978-0-470-38551-7.
Hinkelmann, Klaus and Kempthorne, Oscar (2008). Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (Second ed.). Wiley. ISBN 978-0-471-72756-9.
Hinkelmann, Klaus and Kempthorne, Oscar (2005). Design and Analysis of Experiments, Volume 2: Advanced Experimental Design (First ed.). Wiley. ISBN 978-0-471-55177-5.
^
Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.). New York: Dover. ISBN 0486656853.
Raghavarao, Damaraju and Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific. ISBN 9812563601.
Shah, Kirti R.; Sinha, Bikas K. (1989). "4 Row-Column Designs". Theory of Optimal Designs. Lecture Notes in Statistics. 54. Springer-Verlag. pp. 66–84. ISBN 0-387-96991-8. MR1016151.
Shah, K. R.; Sinha, Bikas K. (1996). "Row-column designs". In S. Ghosh and C. R. Rao. Design and analysis of experiments. Handbook of Statistics. 13. Amsterdam: North-Holland Publishing Co.. pp. 903–937. ISBN 0-444-82061-2. MR1492586.
Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P. [Clarendon]. pp. 400+xiv. ISBN 0198532563.
^ C.J. Colbourn, T. Kløve, and A.C.H. Ling, Permutation arrays for powerline communication, IEEE Trans. Inform. Theory, vol. 50, pp. 1289–1291, 2004.
^ Euler's revolution, New Scientist, 24th of March 2007, pp 48–51
^ Sophie Huczynska, Powerline communication and the 36 officers problem, Philosophical Transactions of the Royal Society A, vol 364, p 3199.
^ C. Colbourn (1984). "The complexity of completing partial latin squares". Discrete Applied Mathematics 8: 25–30. doi:10.1016/0166-218X(84)90075-1.
^ [1]
^ The International Biometric Society

References

Bailey, R.A. (2008). "6 Row-Column designs and 9 More about Latin squares". Design of Comparative Experiments. Cambridge University Press. ISBN 978-0-521-68357-9. MR2422352. Pre-publication chapters are available on-line.
Dénes, J.; Keedwell, A. D. (1974). Latin squares and their applications. New York-London: Academic Press. pp. 547. ISBN 012209350X. MR351850.
Dénes, J. H.; Keedwell, A. D. (1991). Latin squares: New developments in the theory and applications. Annals of Discrete Mathematics. 46. Amsterdam: Academic Press. pp. xiv+454. ISBN 0444888993. MR1096296.
Hinkelmann, Klaus; Kempthorne, Oscar (2008). Design and Analysis of Experiments. I , II (Second ed.). Wiley. ISBN 978-0-470-38551-7. MR2363107.
Hinkelmann, Klaus; Kempthorne, Oscar (2008). Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (Second ed.). Wiley. ISBN 978-0-471-72756-9. MR2363107.
Hinkelmann, Klaus; Kempthorne, Oscar (2005). Design and Analysis of Experiments, Volume 2: Advanced Experimental Design (First ed.). Wiley. ISBN 978-0-471-55177-5. MR2129060.
Knuth, Donald (2011). Volume 4A: Combinatorial Algorithms, Part 1. The Art of Computer Programming (First ed.). Reading, Massachusetts: Addison-Wesley. pp. xv+883pp. ISBN 0-201-03804-8.
Laywine, Charles F.; Mullen, Gary L. (1998). Discrete mathematics using Latin squares. Wiley-Interscience Series in Discrete Mathematics and Optimization. New York: John Wiley & Sons, Inc.. pp. xviii+305. ISBN 0-471-24064-8. MR1644242.
Shah, Kirti R.; Sinha, Bikas K. (1989). "4 Row-Column Designs". Theory of Optimal Designs. Lecture Notes in Statistics. 54. Springer-Verlag. pp. 66–84. ISBN 0-387-96991-8. MR1016151.
Shah, K. R.; Sinha, Bikas K. (1996). "Row-column designs". In S. Ghosh and C. R. Rao. Design and analysis of experiments. Handbook of Statistics. 13. Amsterdam: North-Holland Publishing Co.. pp. 903–937. ISBN 0-444-82061-2. MR1492586.
Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.). New York: Dover. ISBN 0486656853. MR1102899.
Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. New York: Oxford University Press. pp. 400+xiv pp.. ISBN 0198532563, 0-19-853255-5. MR908490.
J. H. van Lint, R. M. Wilson: A Course in Combinatorics. Cambridge University Press 1992,ISBN 0-521-42260-4, p. 157

External links

Latin Squares in the Encyclopaedia of Mathematics
Latin Squares in Java at cut-the-knot
Infinite Latin Square (Java) at cut-the-knot
Magic Square in Latin Square