Latin square#Transversals and rainbow matchings

The Korean mathematician Choi Seok-jeong was the first to publish an example of Latin squares of order nine, in order to construct a magic square in 1700, predating Leonhard Euler by 67 years.{{cite book|last1=Colbourn|first1=Charles J.|last2=Dinitz|first2=Jeffrey H.|title=Handbook of Combinatorial Designs|date=2 November 2006|edition=2nd |publisher=CRC Press|isbn=9781420010541|page=12|url=https://books.google.com/books?id=Q9jLBQAAQBAJ&pg=PA12|access-date=28 March 2017|language=en}}

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.{{harvnb|Dénes|Keedwell|1974|loc=p. 128}} 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:

This Latin square is reduced; both its first row and its first column are alphabetically ordered A, B, C.

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 Latin square

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. Orthogonal arrays are usually written in array form where the triples are the rows, such as:

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 of 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.

{{see also|Small Latin squares and quasigroups}}

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

If we permute the rows, permute the columns, or 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 (that is, permute the three columns in the array form), 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.{{harvnb|Dénes|Keedwell|1974|loc=p. 126}}

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 six isotopy classes.

There is no known easily computable formula for the number {{math|L_n}} of {{math|n × n}} Latin squares with symbols {{math|1, 2, ..., n}}. The most accurate upper and lower bounds known for large {{mvar|n}} are far apart. One classic result{{harvnb|van Lint|Wilson|1992|loc=pp. 161-162}} is that

$$\backslash prod\_\{k=1\}^n\; \backslash left(k!\backslash right)^\{n/k\}\backslash geq\; L\_n\backslash geq\backslash frac\{\backslash left(n!\backslash right)^\{2n\}\}\{n^\{n^2\}\}.$$

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 {{math|L_n}} of {{math|n × n}} Latin squares is

$$L\_n\; =\; n!\; \backslash sum\_\{A\; \backslash in\; B\_n\}^\{\}\; (-1)^\{\backslash sigma\_0\; (A)\}\; \backslash binom\{\backslash operatorname\{per\}\; A\}\{n\},$$

where {{math|B_n}} is the set of all {{math|n × n}} {0, 1}-matrices, {{math|σ₀(A)}} is the number of zero entries in matrix {{mvar|A}}, and {{math|per(A)}} is the permanent of matrix {{mvar|A}}.{{Cite journal|title = A formula for the number of Latin squares|journal = Discrete Mathematics|author1 = Jia-yu Shao|author2 = Wan-di Wei |volume = 110| year = 1992 |issue = 1–3| pages = 293–296|doi=10.1016/0012-365x(92)90722-r|doi-access = free}}

The table below contains all known exact values. It can be seen that the numbers grow exceedingly quickly. For each {{mvar|n}}, the number of Latin squares altogether {{OEIS|id=A002860}} is {{math|n! (n − 1)!}} times the number of reduced Latin squares {{OEIS|id=A000315}}.

For each {{mvar|n}}, each isotopy class {{OEIS|id=A040082}} contains up to {{math|(n!)³}} Latin squares (the exact number varies), while each main class {{OEIS|id=A003090}} contains either 1, 2, 3 or 6 isotopy classes.

The number of structurally distinct Latin squares (i.e. the squares cannot be made identical by means of rotation, reflection, and permutation of the symbols) for {{mvar|n}} = 1 up to 7 is 1, 1, 1, 12, 192, 145164, 1524901344 respectively {{OEIS| id=A264603}}.

{{main|Small Latin squares and quasigroups}}

We give one example of a Latin square from each main class up to order five.

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

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

{{main|Mutually orthogonal Latin squares}}

Two Latin squares of the same order n are called orthogonal if, by overlaying them, one gets every ordered pair (a,b) of symbols where a is a symbol in the first square and b is one in the second square. Orthogonal pairs and more generally sets of pairwise orthogonal Latin squares are important in design theory and finite geometry.

{{See also|Hall-type theorems for hypergraphs#More conditions from rainbow matchings}}

A transversal in a Latin square is a choice of n cells, where each row contains one cell, each column contains one cell, and there is one cell containing each symbol.

One can consider a Latin square as a complete bipartite graph in which the rows are vertices of one part, the columns are vertices of the other part, each cell is an edge (between its row and its column), and the symbols are colors. The rules of the Latin squares imply that this is a proper edge coloring. With this definition, a Latin transversal is a matching in which each edge has a different color; such a matching is called a rainbow matching.

Therefore, many results on Latin squares/rectangles are contained in papers with the term "rainbow matching" in their title, and vice versa.{{cite arXiv|eprint=1208.5670|class=math. CO|first1=Andras|last1=Gyarfas|first2=Gabor N.|last2=Sarkozy|title=Rainbow matchings and partial transversals of Latin squares|year=2012}}

Some Latin squares have no transversal. For example, when n is even, an n-by-n Latin square in which the value of cell i,j is (i+j) mod n has no transversal. Here are two examples:$$$$

\begin{bmatrix}

1 & 2 \\

2 & 1

\end{bmatrix}

\quad

\begin{bmatrix}

1 & 2 & 3 & 4 \\

2 & 3 & 4 & 1 \\

3 & 4 & 1 & 2 \\

4 & 1 & 2 & 3

\end{bmatrix}

In 1967, H. J. Ryser conjectured that, when n is odd, every n-by-n Latin square has a transversal.{{Cite journal|last1=Aharoni|first1=Ron|last2=Berger|first2=Eli|last3=Kotlar|first3=Dani|last4=Ziv|first4=Ran|date=2017-01-04|title=On a conjecture of Stein|journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg|volume=87|issue=2|pages=203–211|doi=10.1007/s12188-016-0160-3|s2cid=119139740|issn=0025-5858}}

In 1975, S. K. Stein and Brualdi conjectured that, when n is even, every n-by-n Latin square has a partial transversal of size n−1.{{Cite journal|last=Stein|first=Sherman|date=1975-08-01|title=Transversals of Latin squares and their generalizations| journal=Pacific Journal of Mathematics| volume=59| issue=2| pages=567–575| doi=10.2140/pjm.1975.59.567|issn=0030-8730|doi-access=free}}

A more general conjecture of Stein is that a transversal of size n−1 exists not only in Latin squares but also in any n-by-n array of n symbols, as long as each symbol appears exactly n times.

Some weaker versions of these conjectures have been proved:

• Every n-by-n Latin square has a partial transversal of size 2n/3.{{Cite journal|last=Koksma|first=Klaas K.| date=1969-07-01|title=A lower bound for the order of a partial transversal in a latin square|journal=Journal of Combinatorial Theory|volume=7|issue=1|pages=94–95|doi=10.1016/s0021-9800(69)80009-8|issn=0021-9800|doi-access=free}}
• Every n-by-n Latin square has a partial transversal of size n − sqrt(n).{{Cite journal| last=Woolbright| first=David E|date=1978-03-01|title=An n × n Latin square has a transversal with at least n−n distinct symbols|journal=Journal of Combinatorial Theory, Series A|volume=24|issue=2|pages=235–237|doi=10.1016/0097-3165(78)90009-2|issn=0097-3165|doi-access=free}}
• Every n-by-n Latin square has a partial transversal of size n − 11 log{{su|b=2|p=2}}(n).{{Cite journal|last1=Hatami|first1=Pooya|last2=Shor|first2=Peter W.|date=2008-10-01|title=A lower bound for the length of a partial transversal in a Latin square|journal=Journal of Combinatorial Theory, Series A|volume=115| issue=7| pages=1103–1113| doi=10.1016/j.jcta.2008.01.002|issn=0097-3165|doi-access=free}}
• Every n-by-n Latin square has a partial transversal of size n − O(log n/loglog n).{{Cite journal |last1=Keevash |first1=Peter |last2=Pokrovskiy |first2=Alexey |last3=Sudakov |first3=Benny |last4=Yepremyan |first4=Liana |date=2022-04-15 |title=New bounds for Ryser's conjecture and related problems |url=https://www.ams.org/btran/2022-09-08/S2330-0000-2022-00092-3/ |journal=Transactions of the American Mathematical Society, Series B |language=en |volume=9 |issue=8 |pages=288–321 |doi=10.1090/btran/92 |issn=2330-0000|doi-access=free |hdl=20.500.11850/592212 |hdl-access=free }}
• Every large enough n-by-n Latin square has a partial transversal of size n −1.{{Cite arXiv |last=Montgomery |first=Richard |date=2023 |title=A proof of the Ryser-Brualdi-Stein conjecture for large even n |class=math.CO |eprint=2310.19779}} (Preprint)

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.{{Cite journal | doi = 10.1002/(sici)1520-6610(1996)4:6<405::aid-jcd3>3.0.co;2-j| title = Generating uniformly distributed random latin squares| journal = Journal of Combinatorial Designs| volume = 4| issue = 6| pages = 405–437| year = 1996| last1 = Jacobson | first1 = M. T. | last2 = Matthews | first2 = P. }}

• In the design of experiments, Latin squares are a special case of row-column designs for two blocking factors.{{citation |author-link=Rosemary A. Bailey|first=R.A.|last=Bailey|chapter=6 Row-Column designs and 9 More about Latin squares|title=Design of Comparative Experiments|publisher=Cambridge University Press|year=2008 |isbn=978-0-521-68357-9|chapter-url=http://www.maths.qmul.ac.uk/~rab/DOEbook|mr=2422352}}{{citation |last1=Shah|first1= Kirti R.|last2=Sinha|first2= Bikas K.| chapter=4 Row-Column Designs|title=Theory of Optimal Designs |series=Lecture Notes in Statistics| volume=54 | publisher=Springer-Verlag | year=1989 | pages=66–84 |isbn=0-387-96991-8 |mr=1016151}}
• In algebra, Latin squares are related to generalizations of groups; in particular, 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.

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.{{cite journal | last1 = Colbourn | first1 = C.J. | author-link = Charles Colbourn | last2 = Kløve | first2 = T. | last3 = Ling | first3 = A.C.H. | year = 2004 | title = Permutation arrays for powerline communication | journal = IEEE Trans. Inf. Theory | volume = 50 | pages = 1289–1291 | doi=10.1109/tit.2004.828150| s2cid = 15920471 }}Euler's revolution, New Scientist, 24 March 2007, pp 48–51{{cite journal | last1 = Huczynska | first1 = Sophie | year = 2006| title = Powerline communication and the 36 officers problem | journal = Philosophical Transactions of the Royal Society A | volume = 364 | issue = 1849| pages = 3199–3214 | doi=10.1098/rsta.2006.1885| pmid = 17090455 | bibcode = 2006RSPTA.364.3199H | s2cid = 17662664 }}

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.

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:

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:

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.

File:Ramanujan_magic_square_construction.svg from a 4×4 Latin square with distinct diagonals and day (D), month (M), century (C) and year (Y) values, and Ramanujan's birthday example]]

The problem of determining if a partially filled square can be completed to form a Latin square is NP-complete.{{cite journal | author = C. Colbourn | author-link = Charles Colbourn | title = The complexity of completing partial latin squares | journal = Discrete Applied Mathematics | volume = 8 | pages = 25–30 | year = 1984 | doi = 10.1016/0166-218X(84)90075-1| doi-access = free }}

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). See also Mathematics of Sudoku.

The more recent KenKen and Strimko puzzles are also examples of Latin squares.

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

Latin squares are used in the design of agronomic research experiments to minimise experimental errors.[http://joas.agrif.bg.ac.rs/archive/article/59 The application of Latin square in agronomic research]

The Latin square also figures in the arms of the Statistical Society of Canada,{{cite web|url=http://www.ssc.ca/archive/main/about/history/arms_e.html|archive-url=https://web.archive.org/web/20130521075715/http://www.ssc.ca/archive/main/about/history/arms_e.html|url-status=dead|archive-date=2013-05-21|title=Letters Patent Confering the SSC Arms|work=ssc.ca}} being specifically mentioned in its blazon. Also, it appears in the logo of the International Biometric Society.[http://www.tibs.org The International Biometric Society] {{webarchive| url=https://web.archive.org/web/20050507083541/http://www.tibs.org/ |date=2005-05-07 }}

File:Latin_cube_order_4.svg

• A Latin rectangle is a generalization of a Latin square in which there are n columns and n possible values, but the number of rows may be smaller than n. Each value still appears at most once in each row and column.
• A Graeco-Latin square is a pair of two Latin squares such that, when one is laid on top of the other, each ordered pair of symbols appears exactly once.
• A Latin hypercube is a generalization of a Latin square from two dimensions to multiple dimensions.

• Block design
• Combinatorial design
• Eight queens puzzle
• Futoshiki
• Magic square
• Problems in Latin squares
• Rook's graph, a graph that has Latin squares as its colorings
• Sator Square
• Vedic square
• Word square

{{reflist}}

• {{cite book |author-link=Rosemary A. Bailey|first=R.A.|last=Bailey|chapter=6 Row-Column designs and 9 More about Latin squares|title=Design of Comparative Experiments|publisher=Cambridge University Press|year=2008 |isbn=978-0-521-68357-9|chapter-url=http://www.maths.qmul.ac.uk/~rab/DOEbook|mr=2422352}}
• {{cite book|last1=Dénes|first1=J.|last2=Keedwell|first2=A. D.|title=Latin squares and their applications|publisher=Academic Press |location=New York-London|year=1974|pages=547|mr=351850|isbn=0-12-209350-X}}
• {{cite book |author1=Shah, Kirti R.|author2=Sinha, Bikas K.| chapter=4 Row-Column Designs|title=Theory of Optimal Designs |series=Lecture Notes in Statistics| volume=54 | publisher=Springer-Verlag | year=1989 | pages=66–84 |isbn=0-387-96991-8 |mr=1016151}}
• {{cite book|first1=J. H.|last1=van Lint|author2-link=Richard M. Wilson|first2=R. M.|last2=Wilson|title=A Course in Combinatorics|url=https://archive.org/details/coursecombinator00lint_417|url-access=limited|publisher=Cambridge University Press|year=1992|isbn=0-521-42260-4|page=[https://archive.org/details/coursecombinator00lint_417/page/n168 157]}}

• {{cite book|last1=Dénes|first1=J. H.|last2=Keedwell|first2=A. D.|title=Latin squares: New developments in the theory and applications|others=Paul Erdős (foreword)|series=Annals of Discrete Mathematics|volume=46|publisher=Academic Press |location=Amsterdam|year=1991 |mr=1096296|isbn=0-444-88899-3}}
• {{cite book | last1=Hinkelmann | first1=Klaus | author-link2=Oscar Kempthorne | last2=Kempthorne | first2=Oscar | year=2008 | title=Design and Analysis of Experiments | volume=I, II |edition=Second | publisher=Wiley |url=http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470385510.html | isbn=978-0-470-38551-7 | mr=2363107}}
• {{cite book |last1=Hinkelmann | first1=Klaus | author-link2=Oscar Kempthorne | last2=Kempthorne | first2=Oscar | year=2008 | title=Design and Analysis of Experiments, Volume I: Introduction to Experimental Design | url=https://books.google.com/books?id=T3wWj2kVYZgC | edition=Second|publisher= Wiley | isbn=978-0-471-72756-9 |mr=2363107}}
• {{cite book |last1=Hinkelmann | first1=Klaus | author-link2=Oscar Kempthorne | last2=Kempthorne | first2=Oscar |year=2005| title=Design and Analysis of Experiments, Volume 2: Advanced Experimental Design|url=https://books.google.com/books?id=GiYc5nRVKf8C|edition=First|publisher= Wiley |isbn=978-0-471-55177-5|mr=2129060}}
• {{cite book|first=Donald|last=Knuth|author-link=Donald Knuth|title=The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1|title-link=The Art of Computer Programming|location=Reading, Massachusetts|publisher=Addison-Wesley| year=2011| isbn=978-0-201-03804-0}}
• {{cite book|last1=Laywine|first1=Charles F.|last2=Mullen|first2=Gary L.| title=Discrete mathematics using Latin squares| series=Wiley-Interscience Series in Discrete Mathematics and Optimization|publisher=John Wiley & Sons, Inc.|location=New York| year=1998|isbn=0-471-24064-8|mr=1644242}}
• {{cite book|author1=Shah, K. R.|author2=Sinha, Bikas K.|chapter=Row-column designs |pages=903–937 | title=Design and analysis of experiments | editor=S. Ghosh and C. R. Rao | series=Handbook of Statistics|volume=13|publisher=North-Holland Publishing Co.|location=Amsterdam|year=1996|isbn=0-444-82061-2|mr=1492586}}
• {{cite book|title=Constructions and Combinatorial Problems in Design of Experiments|author=Raghavarao, Damaraju|location=New York|year=1988|edition=corrected reprint of the 1971 Wiley|publisher=Dover|mr=1102899|isbn=0-486-65685-3|url-access=registration| url=https://archive.org/details/constructionscom0000ragh|author-link=Damaraju Raghavarao}}
• {{cite book|last1=Street|first1=Anne Penfold |author1-link= Anne Penfold Street |last2=Street|first2= Deborah J.|author2-link= Deborah Street |title=Combinatorics of Experimental Design|title-link= Combinatorics of Experimental Design |publisher=Oxford University Press|year=1987|location=New York|isbn=0-19-853256-3| mr=908490 }}
• {{cite book |last1=Berger |first1=Paul D. |last2=Maurer |first2=Robert E. |last3=Celli |first3=Giovana B. |title=Experimental Design with Applications in Management, Engineering, and the Sciences |date=November 28, 2017 |publisher=Springer |pages=267–282 |edition=2nd edition (November 28, 2017) }}

• {{MathWorld |title=Latin Square|id=LatinSquare}}
• [https://encyclopediaofmath.org/wiki/Latin_square Latin Squares] in the Encyclopaedia of Mathematics
• [https://oeis.org/wiki/Index_to_OEIS:_Section_La#Latin Latin Squares] in the Online Encyclopedia of Integer Sequences

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