Latin square

Displaying a 7 × 7 Latin square, this stained glass window honors Ronald Fisher, whose Design of Experiments discussed Latin squares. Fisher's student, A. W. F. Edwards, designed this window for Caius College, Cambridge.

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" was inspired by mathematical papers by Leonhard Euler, who used Latin characters as symbols.^[1] 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 Latin square above 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 (that is, reordering) the rows and columns. Here switching the above matrix's second and third rows yields the following square:

A	B	C
B	C	A
C	A	B

This Latin square 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 following Latin square is:

1	2	3
2	3	1
3	1	2

{ (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^[2] is that

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

A simple and explicit formula for the number of Latin squares was published in 1992, but it is still not easily computable due to the exponential increase in the number of terms.This formula for the number L_n of n × n Latin squares is,^[3]

$L_n = n! \sum_{A \in B_n}^{} (-1)^{\sigma_0 (A)} (^{per A} _{\ \ n}),$

where B_n is the set of all {0,1} n × n matrices, σ₀(A) is the number of zero entries in matrix A, and per(A) is the permanent of matrix A.

The table below contains all 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
1	1	1
2	1	2
3	1	12
4	4	576
5	56	161,280
6	9,408	812,851,200
7	16,942,080	61,479,419,904,000
8	535,281,401,856	108,776,032,459,082,956,800
9	377,597,570,964,258,816	5,524,751,496,156,892,842,531,225,600
10	7,580,721,483,160,132,811,489,280	9,982,437,658,213,039,871,725,064,756,920,320,000
11	5,363,937,773,277,371,298,119,673,540,771,840	776,966,836,171,770,144,107,444,346,734,230,682,311,065,600,000

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
1	1	1
2	1	1
3	1	1
4	2	2
5	2	2
6	12	22
7	147	564
8	283,657	1,676,267
9	19,270,853,541	115,618,721,533
10	34,817,397,894,749,939	208,904,371,354,363,006
11	2,036,029,552,582,883,134,196,099	12,216,177,315,369,229,261,482,540

The number of structurally distinct Latin squares (i.e. the squares cannot be made identical by means of rotation, reflexion, and/or permutation of the symbols) for n = 1 up to 6 is 1, 1, 1, 12, 192, 145164 respectively (sequence A264603 in OEIS) .

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.

Algorithms

For small squares it is possible to generate permutations and test whether the Latin square property is met. For larger squares, Jacobson and Matthews' algorithm allows sampling from a uniform distribution over the space of n × n Latin squares.^[4]

Applications

Statistics and mathematics

• In the design of experiments, Latin squares are a special case of row-column designs for two blocking factors:^[5] Many row-column designs are constructed by concatenating Latin squares.^[6]
• In algebra, Latin squares are generalizations of groups; in fact, Latin squares are characterized as being the multiplication tables (Cayley tables) of quasigroups. A binary operation whose table of values forms a Latin square is said to obey the Latin square property.

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.^[7][8][9]

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.^[10]

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.

Boardgames

Latin squares have been used as the basis for several board games, notably the popular abstract strategy game Kamisado.

Heraldry

The Latin square also figures in the arms of the Statistical Society of Canada,^[11] being specifically mentioned in its blazon. Also, it appears in the logo of the International Biometric Society.^[12]

See also

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

Notes

1. Wallis, W. D.; George, J. C. (2011), Introduction to Combinatorics, CRC Press, p. 212, ISBN 978-1-4398-0623-4.
2. van Lint & Wilson 1992, pp. 161-162
3. "A formula for the number of Latin squares" (PDF). ac.els-cdn.com. Retrieved 2015-09-10.
4. Jacobson, M. T.; Matthews, P. (1996). "Generating uniformly distributed random latin squares". Journal of Combinatorial Designs 4 (6): 405–437. doi:10.1002/(sici)1520-6610(1996)4:6<405::aid-jcd3>3.0.co;2-j.
5. • 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. MR 2422352. 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. External link in |publisher= (help)
     • 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. External link in |publisher= (help)
     • 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. External link in |publisher= (help)
6. • Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.). New York: Dover. ISBN 0-486-65685-3.
   • Raghavarao, Damaraju and Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific. ISBN 981-256-360-1.
   • 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. MR 1016151.
   • 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. MR 1492586.
   • Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P. [Clarendon]. pp. 400+xiv. ISBN 0-19-853256-3.
7. Colbourn, C.J.; Kløve, T.; Ling, A.C.H. (2004). "Permutation arrays for powerline communication". IEEE Trans. Inform. Theory 50: 1289–1291. doi:10.1109/tit.2004.828150.
8. Euler's revolution, New Scientist, 24 March 2007, pp 48–51
9. Huczynska, Sophie. "Powerline communication and the 36 officers problem". Philosophical Transactions of the Royal Society A 364: 3199–3214. doi:10.1098/rsta.2006.1885.
10. C. Colbourn (1984). "The complexity of completing partial latin squares". Discrete Applied Mathematics 8: 25–30. doi:10.1016/0166-218X(84)90075-1.
11. "Letters Patent Confering the SSC Arms". ssc.ca.
12. 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. MR 2422352. Pre-publication chapters are available on-line.
• Dénes, J.; Keedwell, A. D. (1974). Latin squares and their applications. New York-London: Academic Press. p. 547. ISBN 0-12-209350-X. MR 351850.
• Dénes, J. H.; Keedwell, A. D. (1991). Latin squares: New developments in the theory and applications. Annals of Discrete Mathematics 46. Paul Erdős (foreword). Amsterdam: Academic Press. pp. xiv+454. ISBN 0-444-88899-3. MR 1096296.
• Hinkelmann, Klaus; Kempthorne, Oscar (2008). Design and Analysis of Experiments. I , II (Second ed.). Wiley. ISBN 978-0-470-38551-7. MR 2363107.
  • 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. MR 2363107. External link in |publisher= (help)
  • Hinkelmann, Klaus; Kempthorne, Oscar (2005). Design and Analysis of Experiments, Volume 2: Advanced Experimental Design (First ed.). Wiley. ISBN 978-0-471-55177-5. MR 2129060. External link in |publisher= (help)
• 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. MR 1644242.
• 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. MR 1016151.
• 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. MR 1492586.
• Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.). New York: Dover. ISBN 0-486-65685-3. MR 1102899.
• Street, Anne Penfold and Street, Deborah J. (1987). Combinatorics of Experimental Design. New York: Oxford University Press. pp. 400+xiv pp. ISBN 0-19-853256-3. MR 908490. ISBN 0-19-853255-5.
• J. H. van Lint, R. M. Wilson: A Course in Combinatorics. Cambridge University Press 1992,ISBN 0-521-42260-4, p. 157

External links

• Weisstein, Eric W., "Latin Square", MathWorld.
• Latin Squares in the Encyclopaedia of Mathematics
• Latin Squares in Java at cut-the-knot
• Infinite Latin Square (Java) at cut-the-knot