Permutation

In Mathematical, the concept of permutation expresses the idea of rearrangement of discernible objects. A permutation of N distinct objects arranged in a certain order, corresponds to a change about succession of these N objects.

The permutation is one of the basic concepts in Combinatoire, i.e. for problems of enumeration and discrete probabilities. It is thus used to define and study the magic square , the Carré Latin, the Sudoku, or the Rubik' S cubes. The permutations are also used to found the Théorie of the groups, that of the determining, to define the general concept of Symétrie, etc

Definition and examples

Definition

A permutation of a unit X is a Bijection of the unit X on itself.

In particular, a permutation of N elements ($n\; \backslash \; in\; \backslash \; mathbb\; N^*$) is a bijection of a whole finished of cardinal N on itself.

Examples

A permutation of the alphabet of 26 letters is a Mot of 26 letters containing each letter once and only one; and it is clear that this definition remains valid for any alphabet of N letters, with words length N .

There are much of different natures (seven hundred and twenties) in which six bells, of various notes, can be sounded the ones after the others. If the bells are numbered from 1 to 6, then each possible order can be represented by a list of 6 numbers, without repetition, such as for example (3,2,6,5,1,4).

In the same way, six books posed on a shelf and numbered from 1 to 6, can be permuted various manners: arrangement alphabetically, alphabetical order reverse, order preferably, or order chosen “randomly”. Each one of these rearrangements can be seen like a bijection of the whole of the six books, or an identical way, a bijection of the unit $\backslash \; \{1,\; 2,\; \backslash \; cdots,\; 6\; \backslash \}$ on itself. Indeed, if the final order of the books is 3,2,6,5,1,4, one can define the application F : “by” thus

1 is replaced by 3 either F (1) is replaced =3

2 is replaced by 2 or F (2) =2

3 is replaced by 6 or F (3) =6

4 is replaced by 5 or F (4) =5

5 is replaced by 1 or F (5) =1

6 is replaced by 4 or F (6) =4

Finally, the actually permuted objects count little: the permutation can be brought back to a permutation of numbers: numbers of the books, or numbers of bells.

Let us suppose that N people is assoient on N chairs different numbered from 1 with N laid out on the same line. We can consider a placement of these N people on the chairs, like a bijection of the whole of N people on itself, indicating the way in which the people are placed the ones compared to the others on the chairs.

A permutation of N elements is also called permutation without repetition of these elements. Let us announce that formerly a permutation was called substitution .

Enumeration of the permutations

That is to say E a whole with N elements. The problem is to count the permutations of E , i.e. the bijections of E in itself. As for the preceding examples, one can always number the elements of E of 1 with N . To count the permutations of E amounts counting all the possible rearrangements of the list, i.e. all N - uplets formed of the figures of 1 with N in a certain order.

It is possible to give a list of all these rearrangements, in the form of an arborescent representation : there is N choice for the first term of the list. Then for each one of these first choices, there is n-1 possibilities for the second choice, N2 for the third, so on. Finally there is N! (Factorial of N ) possible choices to draw up a list. This method makes it possible to enumerate one and only once each permutation.

Theorem

If X is a finished whole of cardinal N ($n\; \backslash \; in\; \backslash \; mathbb\; \{NR\}$), then the unit $\backslash \; mathfrak\; \{S\}\; (X)$ of the permutations of X is finished and card $\backslash \; mathfrak\; \{S\}\; (X)$= N !.

When $n=0$, the result remains still valid, since by convention, there exists only one application of $\backslash \; varnothing$ in $\backslash \; varnothing$ which moreover is bijective.

It is possible more generally to count the '' p '' - arrangements of N elements, or the injective mappings of a whole of cardinal finished p in a whole of cardinal finished N . This number of arrangements notes $A\_n^p$ and the case of the permutations seems the particular case n=p .

Notation of the permutations

That is to say E a finished unit, of N elements. Even if it means to carry out a classification, to permute the elements of E amounts permuting the entireties of 1 with N . The traditional notation of the permutations places the elements which will be permuted in the natural order on a first line, and the images in correspondence, on a second line. For example

is the application $\backslash \; sigma$ defined by

$\backslash \; sigma\; (1)\; =2,\; \backslash \; sigma\; (2)\; =5,\; \backslash \; sigma\; (3)\; =4,\; \backslash \; sigma\; (4)\; =3,\; \backslash \; sigma\; (5)\; =1$,

or schematically

$\backslash \; begin\; \{matrix\}\; 1\; \backslash \; longmapsto\; 2\; \backslash \; \backslash \; 2\; \backslash \; longmapsto\; 5\; \backslash \; \backslash \; 3\; \backslash \; longmapsto\; 4\; \backslash \; \backslash \; 4\; \backslash \; longmapsto\; 3\; \backslash \; \backslash \; 5\; \backslash \; longmapsto\; 1\; \backslash \; end\; \{matrix\}$.

However, the most practical notation is the canonical form. In this form, the preceding permutation is written:

(1 2 5) (3 4)

what means 1 gives 2 (C. - with-D., 2 is the image of 1) which gives 5 which gives 1; 3 gives 4 which gives 3. This writing corresponds to a décompositon in the form of a composition of circular shifts (see below).

Particular permutations

; Identity: If a permutation leaves the first element in the first place, the second element in the second place, and so on, then it does not change the position of the elements at all. As an application, this permutation is the identical application, and it is called identical permutation . It is generally noted E.

; Transposition: A permutation which is satisfied to exchange two elements I and J by leaving all the other unchanged ones is called transposition . One frequently uses a notation reduced to indicate this permutation: (I, J) . It should be noted that with this choice of notations (I, J) = (J, I) .

; Circular shift: More generally, it is possible to define the circular shifts or cycles . The p - cycle associated with the elements a₁,…, a_p sends the element a₁ on a₂ , then a₂ on a₃ etc, and finally a_p on a₁ . All the other elements remain unchanged. Such a cycle usually notes in the form (a₁,…, a_p) . There still (a₁,…, a_p) = (a₂,…, a_p, a₁) .

Algebraic properties

Composition of permutations

The permutations of E are defined like applications of E in E , it is thus possible to define produces composition to them, which notes $\backslash \; circ$ (but this sign is generally omitted). Precisely, for two permutations $\backslash \; sigma$ and $\backslash \; sigma\text{'}$, to apply $\backslash \; sigma\text{'}$ then $\backslash \; sigma$ amounts applying a permutation $\backslash \; sigma$ = \ sigma \ circ \ sigma' called the produced of $\backslash \; sigma$ and $\backslash \; sigma\text{'}$.

The notation of the permutations is well adapted to the calculation of the product of composition. Thus by taking for example

\ sigma'= \ begin {pmatrix} 1 & 2 & 3 & 4 & 5 \ \ 3&5&2&4&1 \ end {pmatrix} The calculation of the product can be presented on three lines. The first and the second line present the effect of the first permutation $\backslash \; sigma\text{'}$, then one makes correspond to the elements of the second line their image by $\backslash \; sigma$.

Maybe, finally, by striping the intermediate line of calculation

$\backslash \; sigma$ = \ sigma \ circ \ sigma'=

\ begin {pmatrix} 1 & 2 & 3 & 4 & 5 \ \ 4&1&5&3&2 \ end {pmatrix}

It is to be recalled that the law of composition is not commutative.

Structure of group

See also: symmetrical Group

Are N distinct elements in a certain order. To apply a permutation σ amounts modifying the order of it. To return to the initial order is also done by a permutation; this one is noted σ ^-1. More generally, this application σ ^-1, is the reciprocal Application of the bijection σ , since to apply σ then σ ^-1 amounts applying the identical permutation. The permutation σ ^-1 is called the reciprocal permutation or permutation reverses σ .

That is to say E an unspecified unit. The unit $\backslash \; mathfrak\; \{S\}\; (E)$ of the permutations of E is a group for the law of composition $\backslash \; circ$, called symmetrical Groupe of E . In the particular case where $E=\; \backslash \; \{1,\; \backslash \; cdots,\; N\; \backslash \}$ with $n\; \backslash \; in\; \backslash \; mathbb\; \{NR\}$, this unit notes $\backslash \; mathfrak\; \{S\}\; \_n$.

If we consider a unit finished $E$ (formed of elements who are not necessarily entireties) of cardinal $n\; \backslash \; in\; \backslash \; mathbb\; N^\; \{*\}$, we can number the elements of E and identify the permutations of the elements of E with the permutations of N first entireties.

formal Demonstration

To number come down to introduce a bijective mapping $f:\; E\; \backslash \; longrightarrow\; \backslash \; \{1,\; 2,\; \backslash \; cdots,\; N\; \backslash \}$. It is then possible to associate with a permutation $\backslash \; sigma$ of $\backslash \; \{1,\; 2,\; \backslash \; cdots,\; N\; \backslash \}$, the permutation of $E$ defined by $f^\; \{-\; 1\}\; \backslash \; circ\; \backslash \; sigma\; \backslash \; circ\; f$. We thus obtain a bijective mapping (bijection) between the whole of the permutations of $\backslash \; \{1,2,\; \backslash \; cdots,\; N\; \backslash \}$ and that of the permutations of $E$. We can then interpret the permutation of $\backslash \; \{1,\; 2,\; \backslash \; cdots,\; N\; \backslash \}$ preceding as the application (bijective) which sends the element $f^\; \{-\; 1\}\; (1)$ on the element $f^\; \{-\; 1\}\; (2)$, the element $f^\; \{-\; 1\}\; (2)$ on the element $f^\; \{-\; 1\}\; (3)$, and so on.

Plus precisely the application which with $\backslash \; sigma$ associates $f^\; \{-\; 1\}\; \backslash \; circ\; \backslash \; sigma\; \backslash \; circ\; f$ is a Isomorphisme of groups.

Decompositions of the permutations

Decomposition in product of transpositions

Any permutation can be broken up into a product of transpositions. For example, that means that one can, by exchanges two to two, to at will modify the order of the charts of a package.

Such a decomposition is not single: one can for example add an exchange of two charts, then the exchange of the two same charts. On the other hand it is shown that the parity of the necessary number of transpositions remainder the same one. This makes it possible to define the parity and the signature of a permutation (see symmetrical Groupe).

A even Permutation is a permutation which can be expressed like the product of an even number of transpositions. An equivalent definition is that its decomposition in disjoined cycles gives an even number (possibly no one) of even cycles. A odd Permutation is a permutation which can be expressed like product of an odd number of transpositions.

The identical permutation is an even permutation because it can be regarded as the product of 0 transposition, according to convention on the Produit vacuum.

Algorithm of decomposition

Here the general stage of the algorithm of decomposition of a permutation σ

• if the permutation is the identity it is produced of 0 transposition.
• if not it is possible to consider the first nonfixed point by σ

$k=\; \backslash \; min\; \backslash \; \{S,\; 1\; \backslash \; Leq\; S\; \backslash \; Leq\; N,\; \backslash \; sigma\; (S)\; \backslash \; neq\; S\; \backslash \}$ Then by calling τ the transposition which exchanges K and σ ( K ), one forms $\backslash \; sigma\_1=\; \backslash \; tau\; \backslash \; circ\; \backslash \; sigma$ and one takes again the algorithm with σ₁.

One thus forms permutations σ₁, σ₂ etc obtained by multiplying σ by a succession of transpositions τ₁, τ₂ etc, until reaching the identical permutation. Then it comes

$(\backslash \; tau\_p\; \backslash \; tau\_\; \{p-1\}\; \backslash \; dowries\; \backslash \; tau\_1)\; \backslash \; sigma=\; \{\backslash \; rm\; Id\}\; \backslash \; qquad\; \backslash \; sigma=\; \backslash \; tau\_1\; \backslash \; tau\_2\; \backslash \; dowries\; \backslash \; tau\_p$

The validity of the algorithm is justified by noticing that the position of the first nonfixed point increases strictly with each stage, until all the points are fixed. The algorithm is shown after the more n-1 operations, since if the n-1 first points are fixed, they all are to it. Thus it is possible to affirm more precisely than any permutation can be written like product of with the more n-1 transpositions.

Decomposition in product of cycles to disjoined supports

Orbits of an element

The orbit of an element according to a permutation σ is the whole of its successive images obtained by repeated applications of σ. Thus by introducing the permutation σ

Element 1 has as images successively 3,5,6 then again 1,3,5 etc the orbit of 1 is thus the {unit 1,3,5,6}. The orbit of 3 is also the unit {1,3,5,6}, but the orbit of 2 is {2,4,7,8}.

More generally, for an unspecified permutation, the orbits are disjoined and form a partition of the unit {1,2,…, N }. In restriction on a given orbit of size p , the permutation behaves like a circular Shift p elements.

Decomposition

To describe the permutation, it is enough to take an element in each orbit and to give the order of succession of its images per iteration of σ. Thus always with the same example, the permutation σ can be written in the form of a succession of the two cycles (1,3,5,6) and (2,4,7,8). The order of the cycles does not import but the order of the elements inside a cycle must be respected until obtaining a complete cycle. Thus, the same permutation can be written for example

$\backslash \; sigma\; =\; (1,3,5,6)\; \backslash \; circ\; (2,4,7,8)\; =\; (2,4,7,8)\; \backslash \; circ\; (1,3,5,6)$

(7,8,2,4) \ circ (6,1,3,5) \;

In this notation one often omits the symbol of composition $\backslash \; circ$ to reduce the writing.

The “canonical” decomposition of a permutation in “product” of cycles is obtained while initially placing more the small number in first position in each cycle and ordering the cycles according to their first element. This notation often omits the fixed points, i.e. the elements which are their own image by the permutation; thus the permutation (1 3) (2) (4 5) is written simply (1 3) (4 5), since a cycle of only one element does not have any effect.

Applications

Many properties of the permutation σ can be read easily on the decomposition in disjoined cycles: if there is p cycles lengths s₁,…, s_p (by counting the fixed points as cycles length 1)

• the signature of σ is the product of the signatures: it is worth (- 1) ^n-p .
• the order of σ (as a member of the symmetrical Group) is the smallest entirety k>0 such as σ ^K is the identity. It is equal to PPCM lengths of the cycles.
• the combined of a permutation $\backslash \; pi$ by a permutation $\backslash \; sigma$ is the permutation $\backslash \; sigma\; \backslash \; circ\; \backslash \; pi\; \backslash \; circ\; \backslash \; sigma^\; \{-\; 1\}$. One can easily calculate this permutation, by replacing each element I in the decomposition in disjoined cycles of $\backslash \; pi$ by $\backslash \; sigma\_i$.

See too

• Combinative
• Permutation with circular repetition
• Shift
• symmetrical Group
• Signature of a permutation
• Matrix of permutation

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