Latin square
latin square, latin square counterbalancingIn 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 An example of a 3x3 Latin square is:
A  B  C 
C  A  B 
B  C  A 
The name "Latin square" was inspired by mathematical papers by Leonhard Euler 1707–1783, who used Latin characters as symbols,[1] but any set of symbols can be used: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3
Contents
 1 Reduced form
 2 Properties
 21 Orthogonal array representation
 22 Equivalence classes of Latin squares
 23 Number
 24 Examples
 3 Algorithms
 4 Applications
 41 Statistics and mathematics
 42 Error correcting codes
 43 Mathematical puzzles
 44 Board games
 45 Agronomic Research
 5 Heraldry
 6 See also
 7 Notes
 8 References
 9 External links
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
Any Latin square can be 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 
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 a set of n2 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
This means that the n2 ordered pairs r,c are all the pairs i,j with 1 ≤ i, j ≤ n , once each The same is true the ordered pairs r,s and the ordered pairs c,s
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 Ln 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
∏ k = 1 n k ! n / k ≥ L n ≥ n ! 2 n n n 2 ^\leftk!\right^\geq L_\geq }}}}}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 Ln of n × n Latin squares is
L n = n ! ∑ A ∈ B n − 1 σ 0 A per A n , =n!\sum _}^1^A} A}},}
where Bn is the set of all n × n matrices, σ0A is the number of zero entries in matrix A, and perA is the permanent of matrix A[3]
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 the OEIS is n! n1! times the number of reduced Latin squares sequence A000315 in the OEIS
The numbers of Latin squares of various sizesn  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 the OEIS contains up to n!3 Latin squares the exact number varies, while each main class sequence A003090 in the OEIS contains either 1, 2, 3 or 6 isotopy classes
Equivalence classes of Latin squaresn  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 ie the squares cannot be made identical by means of rotation, reflexion, and/or permutation of the symbols for n = 1 up to 7 is 1, 1, 1, 12, 192, 145164, 1524901344 respectively sequence A264603 in the OEIS
Examples
We give one example of a Latin square from each main class up to order 5
[ 1 ] [ 1 2 2 1 ] [ 1 2 3 2 3 1 3 1 2 ] 1\end}\quad 1&2\\2&1\end}\quad 1&2&3\\2&3&1\\3&1&2\end}} [ 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 ] [ 1 2 3 4 2 4 1 3 3 1 4 2 4 3 2 1 ] 1&2&3&4\\2&1&4&3\\3&4&1&2\\4&3&2&1\end}\quad 1&2&3&4\\2&4&1&3\\3&1&4&2\\4&3&2&1\end}} [ 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 ] [ 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 ] 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}\quad 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}}They present, respectively, the multiplication tables of the following groups:
 – the trivial 1element group
 Z 2 _} – the binary group
 Z 3 _} – cyclic group of order 3
 Z 2 × Z 2 _\times \mathbb _} – the Klein fourgroup
 Z 4 _} – cyclic group of order 4
 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 rowcolumn designs for two blocking factors:[5] Many rowcolumn 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
A B C D [ 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 ] E F G H [ 1 3 4 2 2 4 3 1 3 1 2 4 4 2 1 3 ] I J K L [ 1 4 2 3 2 3 1 4 3 2 4 1 4 1 3 2 ] A\\B\\C\\D\\\end}1&2&3&4\\2&1&4&3\\3&4&1&2\\4&3&2&1\\\end}\quad E\\F\\G\\H\\\end}1&3&4&2\\2&4&3&1\\3&1&2&4\\4&2&1&3\\\end}\quad I\\J\\K\\L\\\end}1&4&2&3\\2&3&1&4\\3&2&4&1\\4&1&3&2\\\end}}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:
12 12 123 124 12&12&123&124\\\end}}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:
1 2 1234 4 1&2&1234&4\\\end}}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 proven 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 NPcomplete[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
Board games
Latin squares have been used as the basis for several board games, notably the popular abstract strategy game Kamisado
Agronomic Research
Latin squares are used in the design of agronomic research experiments to minimise experimental errors [11]
Heraldry
The Latin square also figures in the arms of the Statistical Society of Canada,[12] being specifically mentioned in its blazon Also, it appears in the logo of the International Biometric Society[13]
See also
 Block design
 Combinatorial design
 Eight queens puzzle
 Futoshiki
 GraecoLatin square
 Latin hypercube sampling
 Magic square
 Sudoku and 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
 ^ Wallis, W D; George, J C 2011, Introduction to Combinatorics, CRC Press, p 212, ISBN 9781439806234
 ^ van Lint & Wilson 1992, pp 161162
 ^ Jiayu Shao; Wandi Wei 1992 "A formula for the number of Latin squares" PDF Discrete Mathematics 110: 293–296 doi:101016/0012365x9290722r Retrieved 20180410
 ^ Jacobson, M T; Matthews, P 1996 "Generating uniformly distributed random latin squares" Journal of Combinatorial Designs 4 6: 405–437 doi:101002/sici1520661019964:6<405::aidjcd3>30co;2j
 ^
 Bailey, RA 2008 "6 RowColumn designs and 9 More about Latin squares" Design of Comparative Experiments Cambridge University Press ISBN 9780521683579 MR 2422352 Prepublication chapters are available online
 Hinkelmann, Klaus and Kempthorne, Oscar 2008 Design and Analysis of Experiments I and II Second ed Wiley ISBN 9780470385517 External link in publisher= helpCS1 maint: Multiple names: authors list link
 Hinkelmann, Klaus and Kempthorne, Oscar 2008 Design and Analysis of Experiments, Volume I: Introduction to Experimental Design Second ed Wiley ISBN 9780471727569 External link in publisher= helpCS1 maint: Multiple names: authors list link
 Hinkelmann, Klaus and Kempthorne, Oscar 2005 Design and Analysis of Experiments, Volume 2: Advanced Experimental Design First ed Wiley ISBN 9780471551775 External link in publisher= helpCS1 maint: Multiple names: authors list link
 ^
 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, LV 2005 Block Designs: Analysis, Combinatorics and Applications World Scientific ISBN 9812563601 CS1 maint: Multiple names: authors list link
 Shah, Kirti R; Sinha, Bikas K 1989 "4 RowColumn Designs" Theory of Optimal Designs Lecture Notes in Statistics 54 SpringerVerlag pp 66–84 ISBN 0387969918 MR 1016151
 Shah, K R; Sinha, Bikas K 1996 "Rowcolumn designs" In S Ghosh and C R Rao Design and analysis of experiments Handbook of Statistics 13 Amsterdam: NorthHolland Publishing Co pp 903–937 ISBN 0444820612 MR 1492586
 Street, Anne Penfold; Street, Deborah J 1987 Combinatorics of Experimental Design Oxford U P pp 400+xiv ISBN 0198532563
 ^ Colbourn, CJ; Kløve, T; Ling, ACH 2004 "Permutation arrays for powerline communication" IEEE Trans Inf Theory 50: 1289–1291 doi:101109/tit2004828150
 ^ Euler's revolution, New Scientist, 24 March 2007, pp 48–51
 ^ Huczynska, Sophie 2006 "Powerline communication and the 36 officers problem" Philosophical Transactions of the Royal Society A 364: 3199–3214 doi:101098/rsta20061885
 ^ C Colbourn 1984 "The complexity of completing partial latin squares" Discrete Applied Mathematics 8: 25–30 doi:101016/0166218X84900751
 ^ http://joasagrifbgacrs/archive/article/59  The application of Latin square in agronomic research
 ^ "Letters Patent Confering the SSC Arms" sscca Archived from the original on 20130521
 ^ The International Biometric Society Archived 20050507 at the Wayback Machine
References
 Bailey, RA 2008 "6 RowColumn designs and 9 More about Latin squares" Design of Comparative Experiments Cambridge University Press ISBN 9780521683579 MR 2422352 Prepublication chapters are available online
 Dénes, J; Keedwell, A D 1974 Latin squares and their applications New YorkLondon: Academic Press p 547 ISBN 012209350X MR 0351850
 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 0444888993 MR 1096296
 Hinkelmann, Klaus; Kempthorne, Oscar 2008 Design and Analysis of Experiments I , II Second ed Wiley ISBN 9780470385517 MR 2363107
 Hinkelmann, Klaus; Kempthorne, Oscar 2008 Design and Analysis of Experiments, Volume I: Introduction to Experimental Design Second ed Wiley ISBN 9780471727569 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 9780471551775 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: AddisonWesley pp xv+883pp ISBN 0201038048
 Laywine, Charles F; Mullen, Gary L 1998 Discrete mathematics using Latin squares WileyInterscience Series in Discrete Mathematics and Optimization New York: John Wiley & Sons, Inc pp xviii+305 ISBN 0471240648 MR 1644242
 Shah, Kirti R; Sinha, Bikas K 1989 "4 RowColumn Designs" Theory of Optimal Designs Lecture Notes in Statistics 54 SpringerVerlag pp 66–84 ISBN 0387969918 MR 1016151
 Shah, K R; Sinha, Bikas K 1996 "Rowcolumn designs" In S Ghosh and C R Rao Design and analysis of experiments Handbook of Statistics 13 Amsterdam: NorthHolland Publishing Co pp 903–937 ISBN 0444820612 MR 1492586
 Raghavarao, Damaraju 1988 Constructions and Combinatorial Problems in Design of Experiments corrected reprint of the 1971 Wiley ed New York: Dover ISBN 0486656853 MR 1102899
 Street, Anne Penfold; Street, Deborah J 1987 Combinatorics of Experimental Design New York: Oxford University Press pp 400+xiv pp ISBN 0198532563 MR 0908490 ISBN 0198532555
 J H van Lint, R M Wilson: A Course in Combinatorics Cambridge University Press 1992, ISBN 0521422604, p 157
External links
 Weisstein, Eric W "Latin Square" MathWorld
 Latin Squares in the Encyclopaedia of Mathematics
 

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