Each of the six rows is a different permutation of three distinct balls
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.[1]
Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, 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 the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.
Permutations are used in almost every branch of mathematics, and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences.
The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n.
Technically, a permutation of a set S is defined as a bijection from S to itself.[2][3] That is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f[s]. For example, the permutation [3, 1, 2] mentioned above is described by the function defined as
.The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition [performing two given rearrangements in succession], which results in another rearrangement. As properties of permutations do not depend on the nature of the set elements, it is often the permutations of the set that are considered for studying permutations.
In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.
In the popular puzzle Rubik's cube invented in 1974 by Ernő Rubik, each turn of the puzzle faces creates a permutation of the surface colors.
History[edit]
Permutations called hexagrams were used in China in the I Ching [Pinyin: Yi Jing] as early as 1000 BC.
Al-Khalil [717–786], an Arab mathematician and cryptographer, wrote the Book of Cryptographic Messages. It contains the first use of permutations and combinations, to list all possible Arabic words with and without vowels.[4]
The rule to determine the number of permutations of n objects was known in Indian culture around 1150 AD. The Lilavati by the Indian mathematician Bhaskara II contains a passage that translates to:
The product of multiplication of the arithmetical series beginning and increasing by unity and continued to the number of places, will be the variations of number with specific figures.[5]
In 1677, Fabian Stedman described factorials when explaining the number of permutations of bells in change ringing. Starting from two bells: "first, two must be admitted to be varied in two ways", which he illustrates by showing 1 2 and 2 1.[6] He then explains that with three bells there are "three times two figures to be produced out of three" which again is illustrated. His explanation involves "cast away 3, and 1.2 will remain; cast away 2, and 1.3 will remain; cast away 1, and 2.3 will remain".[7] He then moves on to four bells and repeats the casting away argument showing that there will be four different sets of three. Effectively, this is a recursive process. He continues with five bells using the "casting away" method and tabulates the resulting 120 combinations.[8] At this point he gives up and remarks:
Now the nature of these methods is such, that the changes on one number comprehends the changes on all lesser numbers, ... insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body;[9]
Stedman widens the consideration of permutations; he goes on to consider the number of permutations of the letters of the alphabet and of horses from a stable of 20.[10]
A first case in which seemingly unrelated mathematical questions were studied with the help of permutations occurred around 1770, when Joseph Louis Lagrange, in the study of polynomial equations, observed that properties of the permutations of the roots of an equation are related to the possibilities to solve it. This line of work ultimately resulted, through the work of Évariste Galois, in Galois theory, which gives a complete description of what is possible and impossible with respect to solving polynomial equations [in one unknown] by radicals. In modern mathematics, there are many similar situations in which understanding a problem requires studying certain permutations related to it.
Permutations without repetitions[edit]
The simplest example of permutations is permutations without repetitions where we consider the number of possible ways of arranging n items into n places. The factorial has special application in defining the number of permutations in a set which does not include repetitions. The number n!, read "n factorial", is precisely the number of ways we can rearrange n things into a new order. For example, if we have three fruits: an orange, apple and pear, we can eat them in the order mentioned, or we can change them [for example, an apple, a pear then an orange]. The exact number of permutations is then . The number gets extremely large as the number of items [n] goes up.
In a similar manner, the number of arrangements of k items from n objects is sometimes called a partial permutation or a k-permutation. It can be written as [which reads "n permute k"], and is equal to the number [also written as ].[11][12]
Definition[edit]
In mathematics texts it is customary to denote permutations using lowercase Greek letters. Commonly, either and , or and are used.[13]
Permutations can be defined as bijections from a set S onto itself. All permutations of a set with n elements form a symmetric group, denoted , where the group operation is function composition. Thus for two permutations, and in the group , the four group axioms hold:
- Closure: If and are in then so is
- Associativity: For any three permutations ,
- Identity: There is an identity permutation, denoted and defined by for all . For any ,
- Invertibility: For every permutation , there exists an inverse permutation , so that
In general, composition of two permutations is not commutative, that is,
As a bijection from a set to itself, a permutation is a function that performs a rearrangement of a set, and is not an arrangement itself. An older and more elementary viewpoint is that permutations are the arrangements themselves. To distinguish between these two, the identifiers active and passive are sometimes prefixed to the term permutation, whereas in older terminology substitutions and permutations are used.[14]
A permutation can be decomposed into one or more disjoint cycles, that is, the orbits, which are found by repeatedly tracing the application of the permutation on some elements. For example, the permutation defined by has a 1-cycle, while the permutation defined by and has a 2-cycle [for details on the syntax, see § Cycle notation below]. In general, a cycle of length k, that is, consisting of k elements, is called a k-cycle.
An element in a 1-cycle is called a fixed point of the permutation. A permutation with no fixed points is called a derangement. 2-cycles are called transpositions; such permutations merely exchange two elements, leaving the others fixed.
Notations[edit]
Since writing permutations elementwise, that is, as piecewise functions, is cumbersome, several notations have been invented to represent them more compactly. Cycle notation is a popular choice for many mathematicians due to its compactness and the fact that it makes a permutation's structure transparent. It is the notation used in this article unless otherwise specified, but other notations are still widely used, especially in application areas.
Two-line notation[edit]
In Cauchy's two-line notation,[15] one lists the elements of S in the first row, and for each one its image below it in the second row. For instance, a particular permutation of the set S = {1, 2, 3, 4, 5} can be written as
this means that σ satisfies σ[1] = 2, σ[2] = 5, σ[3] = 4, σ[4] = 3, and σ[5] = 1. The elements of S may appear in any order in the first row. This permutation could also be written as:
or
One-line notation[edit]
If there is a "natural" order for the elements of S,[a] say , then one uses this for the first row of the two-line notation:
Under this assumption, one may omit the first row and write the permutation in one-line notation as
,that is, as an ordered arrangement of the elements of S.[16][17] Care must be taken to distinguish one-line notation from the cycle notation described below. In mathematics literature, a common usage is to omit parentheses for one-line notation, while using them for cycle notation. The one-line notation is also called the word representation of a permutation.[18] The example above would then be 2 5 4 3 1 since the natural order 1 2 3 4 5 would be assumed for the first row. [It is typical to use commas to separate these entries only if some have two or more digits.] This form is more compact, and is common in elementary combinatorics and computer science. It is especially useful in applications where the elements of S or the permutations are to be compared as larger or smaller.
Cycle notation[edit]
Cycle notation describes the effect of repeatedly applying the permutation on the elements of the set. It expresses the permutation as a product of cycles; since distinct cycles are disjoint, this is referred to as "decomposition into disjoint cycles".
To write down the permutation in cycle notation, one proceeds as follows:
- Write an opening bracket then select an arbitrary element x of and write it down:
- Then trace the orbit of x; that is, write down its values under successive applications of :
- Repeat until the value returns to x and write down a closing parenthesis rather than x:
- Now continue with an element y of S, not yet written down, and proceed in the same way:
- Repeat until all elements of S are written in cycles.
So the permutation 2 5 4 3 1 [in one-line notation] could be written as [125][34] in cycle notation.
While permutations in general do not commute, disjoint cycles do; for example,
In addition, each cycle can be written in different ways, by choosing different starting points; for example,
One may combine these equalities to write the disjoint cycles of a given permutation in many different ways.
1-cycles are often omitted from the cycle notation, provided that the context is clear; for any element x in S not appearing in any cycle, one implicitly assumes .[19] The identity permutation, which consists only of 1-cycles, can be denoted by a single 1-cycle [x], by the number 1,[b] or by id.[20][21]
A convenient feature of cycle notation is that cycle notation of the inverse permutation is given by reversing the order of the elements in the permutation's cycles. For example,
Canonical cycle notation[edit]
In some combinatorial contexts it is useful to fix a certain order for the elements in the cycles and of the [disjoint] cycles themselves. Miklós Bóna calls the following ordering choices the canonical cycle notation:
- in each cycle the largest element is listed first
- the cycles are sorted in increasing order of their first element
For example, [312][54][8][976] is a permutation in canonical cycle notation.[22] The canonical cycle notation does not omit one-cycles.
Richard P. Stanley calls the same choice of representation the "standard representation" of a permutation.[23] and Martin Aigner uses the term "standard form" for the same notion.[18] Sergey Kitaev also uses the "standard form" terminology, but reverses both choices; that is, each cycle lists its least element first and the cycles are sorted in decreasing order of their least, that is, first elements.[24]
Composition of permutations[edit]
There are two ways to denote the composition of two permutations. is the function that maps any element x of the set to . The rightmost permutation is applied to the argument first,[25] because of the way the function application is written.
Since function composition is associative, so is the composition operation on permutations: . Therefore, products of more than two permutations are usually written without adding parentheses to express grouping; they are also usually written without a dot or other sign to indicate composition.
Some authors prefer the leftmost factor acting first,[26][27][28] but to that end permutations must be written to the right of their argument, often as an exponent, where σ acting on x is written xσ; then the product is defined by xσ·π = [xσ]π. However this gives a different rule for multiplying permutations; this article uses the definition where the rightmost permutation is applied first.
Other uses of the term permutation[edit]
The concept of a permutation as an ordered arrangement admits several generalizations that are not permutations, but have been called permutations in the literature.
k-permutations of n[edit]
A weaker meaning of the term permutation, sometimes used in elementary combinatorics texts, designates those ordered arrangements in which no element occurs more than once, but without the requirement of using all the elements from a given set. These are not permutations except in special cases, but are natural generalizations of the ordered arrangement concept. Indeed, this use often involves considering arrangements of a fixed length k of elements taken from a given set of size n, in other words, these k-permutations of n are the different ordered arrangements of a k-element subset of an n-set [sometimes called variations or arrangements in older literature[c]]. These objects are also known as partial permutations or as sequences without repetition, terms that avoid confusion with the other, more common, meaning of "permutation". The number of such -permutations of is denoted variously by such symbols as , , , , or , and its value is given by the product[29]
,which is 0 when k > n, and otherwise is equal to
The product is well defined without the assumption that is a non-negative integer, and is of importance outside combinatorics as well; it is known as the Pochhammer symbol or as the -th falling factorial power of .
This usage of the term permutation is closely related to the term combination. A k-element combination of an n-set S is a k element subset of S, the elements of which are not ordered. By taking all the k element subsets of S and ordering each of them in all possible ways, we obtain all the k-permutations of S. The number of k-combinations of an n-set, C[n,k], is therefore related to the number of k-permutations of n by:
These numbers are also known as binomial coefficients and are denoted by .
Permutations with repetition[edit]
Ordered arrangements of k elements of a set S, where repetition is allowed, are called k-tuples. They have sometimes been referred to as permutations with repetition, although they are not permutations in general. They are also called words over the alphabet S in some contexts. If the set S has n elements, the number of k-tuples over S is There is no restriction on how often an element can appear in an k-tuple, but if restrictions are placed on how often an element can appear, this formula is no longer valid.
Permutations of multisets[edit]
Permutations of multisets
If M is a finite multiset, then a multiset permutation is an ordered arrangement of elements of M in which each element appears a number of times equal exactly to its multiplicity in M. An anagram of a word having some repeated letters is an example of a multiset permutation.[d] If the multiplicities of the elements of M [taken in some order] are , , ..., and their sum [that is, the size of M] is n, then the number of multiset permutations of M is given by the multinomial coefficient,[30]
For example, the number of distinct anagrams of the word MISSISSIPPI is:[31]
.A k-permutation of a multiset M is a sequence of length k of elements of M in which each element appears a number of times less than or equal to its multiplicity in M [an element's repetition number].
Circular permutations[edit]
Permutations, when considered as arrangements, are sometimes referred to as linearly ordered arrangements. In these arrangements there is a first element, a second element, and so on. If, however, the objects are arranged in a circular manner this distinguished ordering no longer exists, that is, there is no "first element" in the arrangement, any element can be considered as the start of the arrangement. The arrangements of objects in a circular manner are called circular permutations.[32][e] These can be formally defined as equivalence classes of ordinary permutations of the objects, for the equivalence relation generated by moving the final element of the linear arrangement to its front.
Two circular permutations are equivalent if one can be rotated into the other [that is, cycled without changing the relative positions of the elements]. The following four circular permutations on four letters are considered to be the same.
1 4 2 3 4 3 2 1 3 4 1 2 2 3 1 4The circular arrangements are to be read counter-clockwise, so the following two are not equivalent since no rotation can bring one to the other.
1 1 4 3 3 4 2 2The number of circular permutations of a set S with n elements is [n – 1]!.
Properties[edit]
The number of permutations of n distinct objects is n!.
The number of n-permutations with k disjoint cycles is the signless Stirling number of the first kind, denoted by c[n, k].[33]
Cycle type[edit]
The cycles [including the fixed points] of a permutation of a set with n elements partition that set; so the lengths of these cycles form an integer partition of n, which is called the cycle type [or sometimes cycle structure or cycle shape] of . There is a "1" in the cycle type for every fixed point of , a "2" for every transposition, and so on. The cycle type of is
This may also be written in a more compact form as [112231]. More precisely, the general form is , where are the numbers of cycles of respective length. The number of permutations of a given cycle type is[34]
.Conjugating permutations[edit]
In general, composing permutations written in cycle notation follows no easily described pattern – the cycles of the composition can be different from those being composed. However the cycle type is preserved in the special case of conjugating a permutation by another permutation , which means forming the product . Here, is the conjugate of by and its cycle notation can be obtained by taking the cycle notation for and applying to all the entries in it.[35] It follows that two permutations are conjugate exactly when they have the same cycle type.
Permutation order[edit]
The order of a permutation is the smallest positive integer m so that . It is the least common multiple of its cycles lengths. For example, the order of is .
Parity of a permutation[edit]
Every permutation of a finite set can be expressed as the product of transpositions.[36] Although many such expressions for a given permutation may exist, either they all contain an even number of transpositions or they all contain an odd number of transpositions. Thus all permutations can be classified as even or odd depending on this number.
This result can be extended so as to assign a sign, written , to each permutation. if is even and if is odd. Then for two permutations and
It follows that
Matrix representation[edit]
A permutation matrix is an n × n matrix that has exactly one entry 1 in each column and in each row, and all other entries are 0. There are several different conventions that one can use to assign a permutation matrix to a permutation of {1, 2, ..., n}. One natural approach is to associate to the permutation σ the matrix whose [i, j] entry is 1 if i = σ[j] and is 0 otherwise. This convention has two attractive properties: first, the product of matrices and of permutations is in the same order, that is, for all permutations σ and π. Second, if represents the standard column vector [the vector with ith entry equal to 1 and all other entries equal to 0], then .
For example, with this convention, the matrix associated to the permutation is and the matrix associated to the permutation is . Then the composition of permutations is , and the corresponding matrix product is
Composition of permutations corresponding to a multiplication of permutation matrices.
It is also common in the literature to find the inverse convention, where a permutation σ is associated to the matrix whose [i, j] entry is 1 if j = σ[i] and is 0 otherwise. In this convention, permutation matrices multiply in the opposite order from permutations, that is, for all permutations σ and π. In this correspondence, permutation matrices act by permuting indices of standard row vectors : one has .
The Cayley table on the right shows these matrices for permutations of 3 elements.
Foata's transition lemma[edit]
There is a relationship between the one-line and the canonical cycle notation. Consider the permutation , in canonical cycle notation, if we erase its cycle parentheses, we obtain the permutation in one-line notation. Foata's transition lemma establishes the nature of this correspondence as a bijection on the set of n-permutations [to itself].[37] Richard P. Stanley calls this correspondence the fundamental bijection.[23]
Let be the parentheses-erasing transformation. The inverse of is a bit less intuitive. Starting with the one-line notation , the first cycle in canonical cycle notation must start with . As long as the subsequent elements are smaller than , we are in the same cycle. The second cycle starts at the smallest index such that . In other words, is larger than everything else to its left, so it is called a left-to-right maximum. Every cycle in the canonical cycle notation starts with a left-to-right maximum.[37]
For example, in the one-line notation , 5 is the first element larger than 3, so the first cycle must be . Then 8 is the next element larger than 5, so the second cycle is . Since 9 is larger than 8, is a cycle by itself. Finally, 9 is larger than all the remaining elements to its right, so the last cycle is .
The six permutations of with their canonical cycle maps are:
As a first corollary, the number of n-permutations with exactly k left-to-right maxima is also equal to the signless Stirling number of the first kind, . Furthermore, Foata's mapping takes an n-permutation with k-weak excedances to an n-permutations with k − 1 ascents.[37] For example, [2][31] = 321 has two weak excedances [at index 1 and 2], whereas f[321] = 231 has one ascent [at index 1; that is, from 2 to 3].
Permutations of totally ordered sets[edit]
In some applications, the elements of the set being permuted will be compared with each other. This requires that the set S has a total order so that any two elements can be compared. The set {1, 2, ..., n} is totally ordered by the usual "≤" relation and so it is the most frequently used set in these applications, but in general, any totally ordered set will do. In these applications, the ordered arrangement view of a permutation is needed to talk about the positions in a permutation.
There are a number of properties that are directly related to the total ordering of S.
Ascents, descents, runs and excedances[edit]
An ascent of a permutation σ of n is any position i