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Latin square

latin square, latin square counterbalancing
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 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! n-1! times the number of reduced Latin squares sequence A000315 in the 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 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 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 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 1-element group
  • Z 2 _} – the binary group
  • Z 3 _} – cyclic group of order 3
  • Z 2 × Z 2 _\times \mathbb _} – the Klein four-group
  • 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 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

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 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

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
  • Graeco-Latin 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

  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. ^ Jia-yu Shao; Wan-di Wei 1992 "A formula for the number of Latin squares" PDF Discrete Mathematics 110: 293–296 doi:101016/0012-365x9290722-r Retrieved 2018-04-10 
  4. ^ Jacobson, M T; Matthews, P 1996 "Generating uniformly distributed random latin squares" Journal of Combinatorial Designs 4 6: 405–437 doi:101002/sici1520-661019964:6<405::aid-jcd3>30co;2-j 
  5. ^
    • Bailey, RA 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= 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 978-0-471-72756-9  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 978-0-471-55177-5  External link in |publisher= helpCS1 maint: Multiple names: authors list link
  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, LV 2005 Block Designs: Analysis, Combinatorics and Applications World Scientific ISBN 981-256-360-1 CS1 maint: Multiple names: authors list link
    • 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; Street, Deborah J 1987 Combinatorics of Experimental Design Oxford U P pp 400+xiv ISBN 0-19-853256-3 
  7. ^ Colbourn, CJ; Kløve, T; Ling, ACH 2004 "Permutation arrays for powerline communication" IEEE Trans Inf Theory 50: 1289–1291 doi:101109/tit2004828150 
  8. ^ Euler's revolution, New Scientist, 24 March 2007, pp 48–51
  9. ^ Huczynska, Sophie 2006 "Powerline communication and the 36 officers problem" Philosophical Transactions of the Royal Society A 364: 3199–3214 doi:101098/rsta20061885 
  10. ^ C Colbourn 1984 "The complexity of completing partial latin squares" Discrete Applied Mathematics 8: 25–30 doi:101016/0166-218X8490075-1 
  11. ^ http://joasagrifbgacrs/archive/article/59 | The application of Latin square in agronomic research
  12. ^ "Letters Patent Confering the SSC Arms" sscca Archived from the original on 2013-05-21 
  13. ^ The International Biometric Society Archived 2005-05-07 at the Wayback Machine

References

  • Bailey, RA 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 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 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; Street, Deborah J 1987 Combinatorics of Experimental Design New York: Oxford University Press pp 400+xiv pp ISBN 0-19-853256-3 MR 0908490  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

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