Definitions

latin-square

Latin square

A Latin square is an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column. Here is an example:

begin{bmatrix}
1 & 2 & 3 
2 & 3 & 1 
3 & 1 & 2 
end{bmatrix}

Latin squares occur as the multiplication tables (Cayley tables) of quasigroups. They have applications in the design of experiments and in error correcting codes.

The name Latin square originates from Leonhard Euler, who used Latin characters as symbols.

A Latin square is said to be reduced (also, normalized or in standard form) if its first row and first column are in natural order. For example, the Latin square above is reduced because both its first row and its first column are 1,2,3 (rather than 3,1,2 or any other order). We can make any Latin square reduced by permuting (reordering) the rows and columns.

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 first Latin square displayed above 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. The definition of a Latin square can be written in terms of orthogonal arrays as follows:

  • There are n2 triples of the form (r,c,s), where 1 ≤ r, c, sn.
  • All of the pairs (r,c) are different, all the pairs (r,s) are different, and all the pairs (c,s) are different.

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.

The number of Latin squares

There is no known easily-computable formula for the number of n × n Latin squares with symbols 1,2,...,n. The most accurate upper and lower bounds known for large n are far apart. Here we will give all the known exact values. It can be seen that the numbers grow exceedingly quickly.

For each n, the number of Latin squares altogether is n! (n-1)! times the number of reduced Latin squares .

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 161280
6 9408 812851200
7 16942080 61479419904000
8 535281401856 108776032459082956800
9 377597570964258816 5524751496156892842531225600
10 7580721483160132811489280 9982437658213039871725064756920320000
11 5363937773277371298119673540771840 776966836171770144107444346734230682311065600000

For each n, each isotopy class contains up to (n!)3 Latin squares (the exact number varies), while each main class 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 283657 1676267
9 19270853541 115618721533
10 34817397894749939 208904371354363006

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

Latin squares and 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.

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.

Latin squares and mathematical puzzles

The problem of determining if a partially filled square can be completed to form a Latin square is NP-complete.

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 3×3 subgroups must also contain the digits 1–9 (in the standard version).

Heraldry

The Latin square also figures in the blazon of the arms of the Statistical Society of Canada. Also, it appears in the logo of the International Biometric Society.

See also

References

External links

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