Editing Latin square (section)

==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 ''n''<sup>2</sup> triples called the [[orthogonal array]] representation of the square.  For example, the orthogonal array representation of the Latin square

{| class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
|-
| 1|| 2 || 3
|-
| 2 || 3 || 1
|-
| 3 || 1 || 2
|}

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:
{|class="wikitable" style="margin-left:auto;margin-right:auto;text-align:center;width:6em;height:6em;table-layout:fixed;"
!''r''
!''c''
!''s''
|-
|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
|}

The definition of a Latin square can be written in terms of orthogonal arrays: 
* A Latin square is a set of ''n''<sup>2</sup> 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 ''n''<sup>2</sup> 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.

===Equivalence classes of Latin squares===
{{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 ''[[Quasigroup#Homotopy and isotopy|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 [[parastrophe]]s) of the original square.<ref name=DK126>{{harvnb|Dénes|Keedwell|1974|loc=p. 126}}</ref>

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''.<ref name=DK126 /> Each main class contains up to six isotopy classes.

=== Number of {{math|''n'' × ''n''}} Latin squares ===

There is no known easily computable formula for the number {{math|''L<sub>n</sub>''}} 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<ref>{{harvnb|van Lint|Wilson|1992|loc=pp. 161-162}}</ref> is that
<math display="block"> \prod_{k=1}^n \left(k!\right)^{n/k}\geq L_n\geq\frac{\left(n!\right)^{2n}}{n^{n^2}}.</math>

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<sub>n</sub>''}} of {{math|''n'' × ''n''}} Latin squares is
<math display="block"> L_n = n! \sum_{A \in B_n}^{} (-1)^{\sigma_0 (A)} \binom{\operatorname{per} A}{n},</math>
where {{math|''B''<sub>''n''</sub>}} is the set of all {{math|''n'' × ''n''}} {0, 1}-matrices, {{math|σ<sub>0</sub>(''A'')}} is the number of zero entries in matrix {{mvar|A}}, and {{math|per(''A'')}} is the [[Permanent (mathematics)|permanent]] of matrix {{mvar|A}}.<ref>{{Cite journal|title = A formula for the number of Latin squares|journal = [[Discrete Mathematics (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}}</ref>

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}}.

{| class="wikitable" align="center" style="text-align: right;"
|+ The numbers of Latin squares of various sizes
! {{mvar|n}} ||align=right| reduced Latin squares of size {{mvar|n}}<br />{{OEIS|id=A000315}}
! align="right" | all Latin squares of size {{mvar|n}}<br />{{OEIS|id=A002860}}
|-
|align=right| 1 ||align=right| 1 ||align=right| 1
|-
|align=right| 2 ||align=right| 1 ||align=right| 2 
|-
|align=right| 3 ||align=right| 1 ||align=right| 12
|-
|align=right| 4 ||align=right| 4 ||align=right| 576
|-
|align=right| 5 ||align=right| 56 ||align=right| 161,280
|-
|align=right| 6 ||align=right| 9,408 ||align=right| 812,851,200
|-
|align=right| 7 ||align=right| 16,942,080 ||align=right| 61,479,419,904,000
|-
|align=right| 8 ||align=right| 535,281,401,856 ||align=right| 108,776,032,459,082,956,800
|-
|align=right| 9 ||align=right| 377,597,570,964,258,816 ||align=right| 5,524,751,496,156,892,842,531,225,600
|-
| 10 ||align=right| 7,580,721,483,160,132,811,489,280 ||align=right| 9,982,437,658,213,039,871,725,064,756,920,320,000
|-
| 11 ||align=right| 5,363,937,773,277,371,298,119,673,540,771,840 ||align=right| 776,966,836,171,770,144,107,444,346,734,230,682,311,065,600,000 
|-
|12
|1.62 × 10<sup>44</sup>
|
|-
|13
|2.51 × 10<sup>56</sup>
|
|-
|14
|2.33 × 10<sup>70</sup>
|
|-
|15
|1.50 × 10<sup>86</sup>
|
|}

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

{| class="wikitable" align="center" style="text-align: right;"
|+ Equivalence classes of Latin squares
! {{mvar|n}} ||align=right| main classes
{{OEIS|id=A003090}} 
! align="right" | isotopy classes
{{OEIS|id=A040082}}
!structurally distinct squares
{{OEIS|id=A264603}}
|-
|align=right| 1 ||align=right| 1 ||align=right| 1
|1
|-
|align=right| 2 ||align=right| 1 ||align=right| 1
|1
|-
|align=right| 3 ||align=right| 1 ||align=right| 1
|1
|-
|align=right| 4 ||align=right| 2 ||align=right| 2
|12
|-
|align=right| 5 ||align=right| 2 ||align=right| 2
|192
|-
|align=right| 6 ||align=right| 12 ||align=right| 22
|145,164
|-
|align=right| 7 ||align=right| 147 ||align=right| 564<!-- Note 564, not 563, see Talk page!-->
|1,524,901,344
|- 
|align=right| 8 ||align=right| 283,657 ||align=right| 1,676,267
|
|- 
|align=right| 9 ||align=right| 19,270,853,541 ||align=right| 115,618,721,533
|
|-
| 10 ||align=right| 34,817,397,894,749,939 ||align=right| 208,904,371,354,363,006
|
|-
| 11 ||align=right| 2,036,029,552,582,883,134,196,099 ||align=right| 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, 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}}.

===Examples===
{{main|Small Latin squares and quasigroups}}
We give one example of a Latin square from each main class up to order five.

<div class="center"><math>
\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}
</math></div>

<div class="center"><math>
\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}
</math></div>

<div class="center"><math>
\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}
</math></div>

They present, respectively, the multiplication tables of the following groups:
*{0} – the trivial 1-element group
*<math>\mathbb{Z}_2</math> – the [[Binary numeral system|binary]] group
*<math>\mathbb{Z}_3</math> – [[cyclic group]] of order 3
*<math>\mathbb{Z}_2 \times \mathbb{Z}_2</math> – the [[Klein four-group]]
*<math>\mathbb{Z}_4</math> – cyclic group of order 4
*<math>\mathbb{Z}_5</math> – cyclic group of order 5
* the last one is an example of a [[quasigroup]], or rather a [[Loop (algebra)|loop]], which is not associative.
<!--- Any comments about the last example? [[User:Ilya Schurov|Ilya Schurov]] 10:23, 23 October 2005 (UTC) --->

===Orthogonal pairs===
{{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.