Permutation

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In mathematics, the idea of ​​ permutations relates to the action of organizing all group members into several sequences or sequences, or if the sets are already ordered, rearranging (reordering ) elements, a process called permuting . This is different from the combination, which is the choice of some of the set members in which the order is ignored. For example, written as a tuple, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2, 3 , 1), (3,1,2), and (3,2,1). These are all possible sequences of these three sets of elements. For another example, the anagram of a word, all different letters, is a permutation of the letters. In this example, the letters are already ordered in the original word and the anagrams are rearranging letters. The study of finite set permutations is a topic in the field of combinatorics.

Permutations occur, in a way more or less prominent, in almost all areas of mathematics. They often appear when different sequences on a particular limited set are considered, probably just because people want to ignore the sequence and need to know how many configurations are identified. For the same reason permutations appear in the study of sorting algorithms in computer science.

The number of permutations of different n objects is n factorial, usually written as n ! , which means the product of all positive integers is less than or equal to n .

In algebra and especially in group theory, the permutations of the set S are defined as a grain of S to self. That is, it is a function of S to S that each element occurs exactly once as the value of the image. This is related to the rearrangement of the S element where each element s > is replaced by the corresponding f ( s ) . The permutation collections form a group called symmetric group S . The key to this group structure is the fact that the composition of two permutations (doing two consecutive rearrangements) results in another rearrangement. Permutations may act on a structured object by rearranging its components, or with certain replacement (replacement) symbols.

In basic combinatorics, k -permutations, or partial permutations, are the ordered settings of k different elements selected from one set. When k

Video Permutation

Histori

The rules for determining the number of permutations of objects n are known in Indian culture at least as early as around 1150: Lilavati by Indian mathematician Bhaskara II contains sections that are translated into

An arithmetic array propagation product that begins and increases with unity and continues into a number of places, will be a variation of numbers with certain numbers.

Fabian Stedman in 1677 explained the factorial when explaining the number of bell permutations in rings. Starting from two bells: "first, two must be received varies in two ways" which he illustrates by showing 1 2 and 2 1. He then explains that with three bells there are "three times two numbers resulting from three "which is again illustrated. The explanation involves "discarded 3, and 1,2 will remain, discarded 2, and 1,3 will remain, discarded 1, and 2.3 will remain". He then moves to four bells and repeats the casting argument which shows that there will be four distinct sets of three. This is effectively a recursive process. He went on with five bells using the "casting away" method and tabulated the resulting 120 combinations. At this point he gave up and commented:

Now the nature of this method is that, that the change in one number understands the change in all smaller numbers,... such that a complement of Peal of change on a number seems to be formed by uniting Compleat Peals on all smaller numbers. into one whole body;

Stedman extended permutation considerations; he went on to consider the number of alphabetic and horizontal letter permutations from a stable of 20.

A first case in which seemingly unrelated mathematical questions studied with the aid of permutations occurred around 1770, when Joseph Louis Lagrange, in the study of polynomial equations, observed that the root permutation of an equation is related to the possibility of completion. This line of work ultimately yields, through the work of ÃÆ'â € variste Galois, in Galois theory, which gives a full description of what is possible and impossible with respect to the solving of polynomial equations (in one unknown) by radicals. In modern mathematics there are many similar situations where understanding the problem requires studying certain permutations associated with it.

Maps Permutation

Definition and notation

There are two common, equivalent ways of permutation, sometimes called active and passive , or in terms of substitutions and permutations . Which form is preferred depends on the type of question asked in the given discipline.

The way is active to assume permutations of the set S (limited or unlimited) is to define it as a prejudice from S to itself. Thus, a permutation is considered a function that can be composed with one another, forming a permutation group. From this point of view, the S element does not have an internal structure and only labels for the moved object: one can refer to permutations of the n set of elements as "permutations on n letter ".

Dalam notasi dua baris Cauchy , satu daftar elemen S di baris pertama, dan untuk masing-masing gambarnya di bawahnya di baris kedua. Misalnya, permutasi khusus dari himpunan S = {1,2,3,4,5} dapat ditulis sebagai:

${\ displaystyle \ sigma = {\ begin {pmatrix} 1 & amp; 2 & amp; 3 & amp; 4 & amp; 5 \\ 2 & amp; 5 & amp; 4 & amp; 3 & amp; 1 \ end {pmatrix}} ;}  ÂÂ$

ini berarti bahwa ? memenuhi ? (1) = 2 , ? (2) = 5 , ? (3) = 4 , ? (4) = 3 , dan ? (5) = 1 . Elemen S dapat muncul dalam urutan apa pun di baris pertama. Permutasi ini juga bisa ditulis sebagai:

${\ displaystyle \ sigma = {\ begin {pmatrix} 3 & amp; 2 & amp; 5 & amp; 1 & amp; 4 \\ 4 & amp; 5 & amp; 1 & amp; 2 & amp; 3 \ end {pmatrix}}.}  ÂÂ$

Di bawah asumsi ini, seseorang dapat menghilangkan baris pertama dan menulis permutasi dalam notasi satu baris sebagai ${\ displaystyle \ sigma (x_ {1}) \; \ sigma (x_ {2}) \; \ sigma (x_ {3}) \; \ cdots \; \ sigma (x_ {n})}  ÂÂ$ , yaitu, pengaturan yang terurut dari S. Care harus diambil untuk membedakan notasi satu baris dari notasi siklus yang dijelaskan kemudian. Dalam literatur matematika, penggunaan umum adalah untuk menghilangkan tanda kurung untuk notasi satu baris, sementara menggunakannya untuk notasi siklus. Notasi satu baris juga disebut representasi kata dari permutasi. Contoh di atas akan menjadi 2 5 4 3 1 karena urutan alami 1 2 3 4 5 akan diasumsikan untuk baris pertama. (Hal ini khas untuk menggunakan koma untuk memisahkan entri ini hanya jika beberapa memiliki dua atau lebih digit.) Formulir ini lebih kompak, dan umum dalam kombinatorik dasar dan ilmu komputer. Ini sangat berguna dalam aplikasi di mana unsur-unsur S atau permutasi harus dibandingkan sebagai lebih besar atau lebih kecil.

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Penggunaan lain dari istilah permutasi

The concept of permutations as ordered arrangements recognizes some non-permutation generalizations but has been called permutations in the literature.

k -permutation of n

yaitu 0 ketika k & gt; n , dan sebaliknya sama dengan

${\ displaystyle {\ frac {n!} {(n-k)!}}.}  ÂÂ$

Produk ini didefinisikan dengan baik tanpa asumsi bahwa ${\ displaystyle n}  ÂÂ$ adalah bilangan bulat non-negatif dan penting juga di luar kombinatorik; ini dikenal sebagai simbol Pochhammer ${\ displaystyle (n) _ {k}}  ÂÂ$ atau sebagai ${\ displaystyle k}  ÂÂ$ -faktor faktorial jatuh ${\ displaystyle n ^ {\ garis bawah {k}}}  ÂÂ$ dari ${\ displaystyle n}  ÂÂ$ .

The use of the term permutation is strongly related to the term combination . The k combination of n isset S its element is not ordered. By taking all the k subset elements S and ordering each of them in all the ways we can get all k -permutation S . The number k -complete from n -set, C ( n , k ) , because it relates to the number of k -pubutation n by:

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂC()\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Ân,Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂkÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â)Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\frac{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂPÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â()\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂnÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â,Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âk}{}}$                                       P               ()               k               ,               k                                       =                                                                               n                   !                                Â

















<                   n                   -                   k                   !                                                                                                       k                   !                                                   0                   !                                                                     =                                            n               !                                       ()              n               -               k               !                             k               !                                       .           {Annotation encoding = "application/x-tex"> {\ displaystyle C (n, k) = {\ frac {P (n, k)} {P (k, k)}} = {\ frac {\ tfrac {n!} {(nk)!}} {\ tfrac {k!} {0!}}} = {\ frac {n!} {(nk)! \, k!}}.}  Â

Angka-angka ini juga dikenal sebagai koefisien binomial dan dilambangkan ${\ displaystyle {\ binom {n} {k}}}  ÂÂ$ .

Permutasi dengan pengulangan

Instructs the setting of elements set S long n where repetitions are allowed called n -tuples, but are sometimes referred to as permutations with repetition though non-permutations in general. They are also called above the alphabet S in some contexts. If the set S has a k element, the number n -tuple above S is:

${\ displaystyle k ^ {n}.}$

There is no limit to how often an element can appear in n -tuple, but if the restriction is placed on how often an element can appear, this formula is no longer valid.

Permutations multisets

Misalnya, jumlah anagram yang berbeda dari kata MISSISSIPPI adalah:

${\ displaystyle {\ frac {11!} {1! \, 4! \, 4! \, 2!}} = 34650}  ÂÂ$ .

k-permutation of multiset M is the long sequence k of the M element in which each element appears most many the number in M times (number repetition a ).

Circular Permutations

Permutations, when considered as settings, are sometimes referred to as a linearly ordered arrangement. In this setting there is the first element, the second element, and so on. However, if the objects are arranged in a circle, this distinguished sequence no longer exists, that is, there is no "first element" in the setting, each element can be regarded as the beginning of the arrangement. Setting objects in a circular way is called circular permutation . This can be formally defined as the equivalence class of the ordinary permutations of the object, for the resulting equality relation by moving the last element of the forward linear arrangement.

Two circular permutations are equivalent if one can be rotated to another (ie, cycling without changing the relative position of the element). The following two circular permutations on four letters are considered equal.

 1 4  Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 4 3 2 1  Â Â Â Â Â Â Â Â Â · Â · 2 Â Â Ã Â Ã, Â Â Â 2 3  

The circular arrangement must be read counterclockwise, so the following two are unequal because no rotation can bring one to the other.

 1 1  Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 4 3 3 4  Â 2 2 2 3 4 5 6 7 8 9 10  

The number of circular permutations of sets S with n elements is ( n - 1)!.

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Permutations in group theory

The set of all permutations of a particular set of S groups, with the map composition as the product operation and the identity function as a neutral element of the group. This is a symmetric group of S , denoted by Sym ( S ). Until the isomorphism, this symmetric group depends only on the cardinality of the set (called degree group), so the properties of the S element are irrelevant to the structure of the group. Symmetrical groups have been studied extensively in the case of finite sets, so, confined to this case, one can assume without losing the, > n } for native numbers n . This is a symmetric group of degrees n , usually written as S _n .

Each subgroup of symmetric groups is called permutation group . According to Cayley's theorem, each group is isomorphic to several groups of permutations, and each group is limited to a subgroup of some finite symmetric groups.

Notation cycle

This alternative notation explains the effect of repeatedly applying permutations, regarded as a function of one set to itself. This reveals permutations as the product of a cycle corresponding to the orbit of the permutation; because of different disjoint orbits, this is referred to as "decomposition into a disjoint cycle".

A cycle ( x ) of length 1 occurs when? ( x ) = x , and is usually omitted from the cycle notation, as long as the set S is clear: for any element x ? S does not appear in one cycle, one implicitly assumes? ( x ) = x . The identity permutation, which consists of only 1-cycle, can be denoted by a 1-cycle (x), with number 1, or with id .

Cycle ( x ₁ x ₂ ... x _k ) of the length k is called k -cycle. Written on its own, it shows a permutation in itself, which maps x _i to x _{i 1} for i & lt; k , and x _k to x ₁ , while implicitly mapping all other elements of S to themselves (eliminated 1-cycle). Therefore, the individual cycle in cycle notation can be interpreted as a factor in a product composition. Because the orbit is disjoint, the associated cycles will change in the composition, and can be written in any order. Decomposition cycle is basically unique: apart from reordering the cycle in the product, there is no other way to write ? as the product of the cycle. Each individual cycle can be written in different ways, as in the example above where (5 1 2) and (1 2 5) and (2 5 1) all show the same cycle, although note that (5 2 1) shows the difference cycle.

In 1-cycle element ( x ), according to? ( x ) = x , is called a fixed point of permutation?. A fixed-point permutation is called chaos. A long cycle of two is called transposition; Such permutations simply swap places of two elements, implicitly leaving the other fixed. Because of the permutation partition orbit set S , for the limited set of size n , the length of the permutation cycle? forming a n partition called a cycle type ?. There is a "1" in the cycle type for each fixed point ?, "2" for each transposition, and so on. Cycle type? = (1 2 5) (3 4) (6 8) (7), is (3,2,2,1) which is sometimes written in a more concise form such as (1 ¹ 2 ² , 3 ¹ ).

Jumlah n -permutation dengan siklus k disjoint adalah nomor Stirling tanpa tanda dari jenis pertama, dilambangkan dengan ${\ displaystyle c (n, k)}  ÂÂ$ .

Grup abstrak vs. permutasi vs. tindakan grup

Permutation groups have more structures than abstract groups, and different realizations of groups as permutation groups need not be equivalent to permutations. For example S ₃ is naturally a permutation group, where each transposition has a cycle type (2,1); but the Cayley theorem proof realizes S ₃ as a subgroup S ₆ (ie the permutation of the 6 elements S ₃ itself), where the transposition of the permutation group has a cycle type (2,2,2 ). Finding a minimum-order symmetric group containing an isomorphic subgroup to a particular group (sometimes called minimal loyalty representation) is a rather difficult problem. Thus apart from Cayley's theorem, the study of permutation groups is different from abstract group studies, being branches of representational theory.

Most permutation forces can return in abstract settings by considering group actions instead. A group action actually defines the elements of a set according to the prescriptions provided by the abstract group. For example, S ₃ act faithfully and transitively on a set with exactly three elements (by allowing it).

Products and inverse

The product of two permutations is defined as their composition as a function, in other words ? A A·? is a function that maps every element x from set to ? (? ( x )). Notice that the rightmost permutation is applied to the first argument, because the way the application function is written. Some authors prefer the leftmost factor acting first, but for that final permutation must be written to right of their argument, for example as an exponent, where ? acting on x written x ^? ; then the product is determined by x ^{? A A·?} = ( x ^? ) ^? . However this gives different rules to multiply permutations; This article uses a definition in which the rightmost permutation is applied first.

Since the composition of the two demands always gives the other ore, the product of the two permutations is a permutation again. In a two-line notation, the product of two permutations is obtained by rearranging the second permutation column (far left) so that the first row is identical to the second row of the first (rightmost) permutation. The product can then be written as the first line of the first permutation above the second row of the modified second permutation. For example, given permutations,

${\ displaystyle P = {\ begin {pmatrix} 1 & amp; 2 & amp; 3 & amp; 4 & amp; 5 \\ 2 & amp; 4 & amp; 1 & amp; 3 & amp; 5 \ end {pmatrix}} \ quad {\ text {and}} \ quad Q = {\ begin {pmatrix} 1 & amp; 2 & amp; 3 & amp; 4 & amp; 5 \\ 5 & amp; 4 & amp; 3 & amp; 2 & amp; 1 \ end {pmatrix}},}$

Dalam notasi siklus, produk yang sama ini akan diberikan oleh:

${\ di$

Source of the article : Wikipedia