Permutation

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.
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 distinct objects is  factorial, usually written as , which means the product of all positive integers less than or equal to .
Technically, a permutation of a set is defined as a bijection from to itself. That is, it is a function from to for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of in which each element is replaced by the corresponding . For example, the permutation (3,1,2) mentioned above is described by the function formula_1 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 formula_3 that are considered for studying permutations. 
In elementary combinatorics, the -permutations, or partial permutations, are the ordered arrangements of distinct elements selected from a set. When is equal to the size of the set, these are the permutations of the set.
History.
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.
The rule to determine the number of permutations of "n" objects was known in Indian culture around 1150. 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.
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. 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". 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. 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;
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.
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.
The simplest example of permutations is permutations without repetitions where we consider the number of possible ways of arranging items into 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 formula_4. The number gets extremely large as the number of items (n) goes up.
In a similar manner, the number of arrangements of r items from n objects is consider a partial permutation. It is written as formula_5 (which reads "n permute r"), and is equal to the number formula_6 (also written as formula_7).
Definition.
In mathematics texts it is customary to denote permutations using lowercase Greek letters. Commonly, either formula_1 and formula_9, or formula_10 and formula_11 are used.
Permutations can be defined as bijections from a set onto itself. All permutations of a set with "n" elements form a symmetric group, denoted formula_12, where the group operation is function composition. Thus for two permutations, formula_11 and formula_14 in the group formula_12, the four group axioms hold: 
In general, composition of two permutations is not commutative, that is, formula_30
As a bijection from a set to itself, a permutation is a function that "performs" a rearrangement of a set, and is not a rearrangement itself. An older and more elementary viewpoint is that permutations are the rearrangements 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.
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 formula_14 defined by formula_32 has a 1-cycle, formula_33 while the permutation formula_11 defined by formula_35 and formula_36 has a 2-cycle formula_37 (for details on the syntax, see 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 formula_38 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.
 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.
In Cauchy's "two-line notation", 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 , , , , and . The elements of "S" may appear in any order in the first row. This permutation could also be written as:
or
One-line notation.
If there is a "natural" order for the elements of "S", say formula_42, 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, an ordered arrangement of S. 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. The example above would then be since the natural order 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.
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 formula_14 in cycle notation, one proceeds as follows:
Since for every new cycle the starting point can be chosen in different ways, there are in general many different cycle notations for the same permutation; for the example above one has:
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 formula_53. The identity permutation, which consists only of 1-cycles, can be denoted by a single 1-cycle (x), by the number 1, or by "id".
A convenient feature of cycle notation is that one can find a permutation's inverse simply by reversing the order of the elements in the permutation's cycles. For example
Canonical cycle notation ( standard form).
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":
For example, (312)(54)(8)(976) is a permutation in canonical cycle notation. The canonical cycle notation does not omit one-cycles.
Richard P. Stanley calls the same choice of representation the "standard representation" of a permutation. and Martin Aigner uses the term "standard form" for the same notion. 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.
Composition of permutations.
There are two ways to denote the composition of two permutations. formula_55 is the function that maps any element "x" of the set to formula_56. The rightmost permutation is applied to the argument first,
because of the way the function application is written. 
Since function composition is associative, so is the composition operation on permutations: formula_57. 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,
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 "x""σ"; then the product is defined by . 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".
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".
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 the older literature). 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 formula_58-permutations of formula_59 is denoted variously by such symbols as formula_60, formula_61, formula_62, formula_63, or formula_64, and its value is given by the product
which is 0 when , and otherwise is equal to
The product is well defined without the assumption that formula_59 is a non-negative integer, and is of importance outside combinatorics as well; it is known as the Pochhammer symbol formula_68 or as the formula_58-th falling factorial power formula_70 of formula_59.
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 formula_73.
Permutations with repetition.
Ordered arrangements of "n" elements of a set "S", where repetition is allowed, are called "n"-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 "k" elements, the number of "n"-tuples over "S" is formula_74
There is no restriction on how often an element can appear in an "n"-tuple, but if restrictions are placed on how often an element can appear, this formula is no longer valid.
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. If the multiplicities of the elements of "M" (taken in some order) are formula_75, formula_76, ..., formula_77 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,
For example, the number of distinct anagrams of the word MISSISSIPPI is:
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.
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. 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 two circular permutations on four letters are considered to be the same.
The circular arrangements are to be read counterclockwise, so the following two are not equivalent since no rotation can bring one to the other.
The number of circular permutations of a set "S" with "n" elements is ("n" – 1)!.
Properties.
The number of permutations of distinct objects is !.
The number of -permutations with disjoint cycles is the signless Stirling number of the first kind, denoted by .
Permutation type.
The cycles of a permutation partition the set formula_12 so the lengths of the cycles of a permutation formula_14 form a partition of "n" called the cycle type of formula_14. There is a "1" in the cycle type for every fixed point of σ, a "2" for every transposition, and so on. The cycle type of formula_83 is (3,2,2,1) which is sometimes written in a more compact form as [112231]. 
The general form is formula_84, where formula_85 are the numbers of cycles of respective length. The number of permutations of a certain type is
Conjugating permutations.
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 structure is preserved in the special case of conjugating a permutation formula_14 by another permutation formula_11, which means forming the product formula_89. Here, formula_89 is the "conjugate" of formula_14 and its cycle notation can be obtained by taking the cycle notation for formula_14 and applying formula_11 to all the entries in it. It follows that two permutations are conjugate exactly when they have the same type.
Permutation order.
The order of a permutation formula_14 is the smallest positive integer "m" so that formula_95. It is the least common multiple of its cycles lengths. For example, the order of formula_96 is formula_97.
Parity of a permutation.
Every permutation of a finite set can be expressed as the product of transpositions.
Although many such expressions for a given permutation may exist, either they all contain an even or 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 formula_98, to each permutation. formula_99 if formula_14 is even and formula_101 if formula_14 is odd. Then for two permutations formula_14 and formula_11
It follows that formula_106
Matrix representation.
One can represent a permutation of {1, 2, ..., "n"} as an "n"×"n" matrix. There are two natural ways to do so, but only one for which multiplications of matrices corresponds to multiplication of permutations in the same order: this is the one that associates to "σ" the matrix "M" whose entry "M""i","j" is 1 if "i" = "σ"("j"), and 0 otherwise. The resulting matrix has exactly one entry 1 in each column and in each row, and is called a "permutation matrix". 
 is a list of these matrices for permutations of 4 elements. The Cayley table on the right shows these matrices for permutations of 3 elements.
Foata's transition lemma.
There is a relationship between the one-line and the canonical cycle notation. Consider the permutation formula_107, in canonical cycle notation, if we erase its cycle parentheses, we obtain the permutation formula_108 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). Richard P. Stanley calls this correspondence the "fundamental bijection". 
Let formula_109 be the parentheses-erasing transformation. The inverse of formula_110 is a bit less intuitive. Starting with the one-line notation formula_111, the first cycle in canonical cycle notation must start with formula_112. As long as the subsequent elements are smaller than formula_112, we are in the same cycle. The second cycle starts at the smallest index formula_114 such that formula_115. In other words, formula_116 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.
For example, in the one-line notation formula_117, 5 is the first element larger than 3, so the first cycle must be formula_118. Then 8 is the next element larger than 5, so the second cycle is formula_119. Since 9 is larger than 8, formula_120 is a cycle by itself. Finally, 9 is larger than all the remaining elements to its right, so the last cycle is formula_121.
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, formula_122. Furthermore, Foata's mapping takes an "n"-permutation with "k"-weak excedances to an "n"-permutations with ascents. For example, (2)(31) = 321 has two weak excedances (at index 1 and 2), whereas has one ascent (at index 1; that is, from 2 to 3).
Permutations of totally ordered sets.
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.
An "ascent" of a permutation "σ" of "n" is any position "i" < "n" where the following value is bigger than the current one. That is, if "σ" = "σ"1"σ"2..."σ""n", then "i" is an ascent if "σ""i" < "σ""i"+1.
For example, the permutation 3452167 has ascents (at positions) 1, 2, 5, and 6.
Similarly, a "descent" is a position "i" < "n" with "σ""i" > "σ""i"+1, so every "i" with formula_123 either is an ascent or is a descent of "σ".
An "ascending run" of a permutation is a nonempty increasing contiguous subsequence of the permutation that cannot be extended at either end; it corresponds to a maximal sequence of successive ascents (the latter may be empty: between two successive descents there is still an ascending run of length 1). By contrast an "increasing subsequence" of a permutation is not necessarily contiguous: it is an increasing sequence of elements obtained from the permutation by omitting the values at some positions.
For example, the permutation 2453167 has the ascending runs 245, 3, and 167, while it has an increasing subsequence 2367.
If a permutation has "k" − 1 descents, then it must be the union of "k" ascending runs.
The number of permutations of "n" with "k" ascents is (by definition) the Eulerian number formula_124; this is also the number of permutations of "n" with "k" descents. Some authors however define the Eulerian number formula_124 as the number of permutations with "k" ascending runs, which corresponds to descents.
An excedance of a permutation "σ"1"σ"2..."σ""n" is an index "j" such that . If the inequality is not strict (that is, ), then "j" is called a "weak excedance". The number of "n"-permutations with "k" excedances coincides with the number of "n"-permutations with "k" descents.
Inversions.
An "inversion" of a permutation "σ" is a pair of positions where the entries of a permutation are in the opposite order: formula_126 and formula_127. So a descent is just an inversion at two adjacent positions. For example, the permutation has three inversions: (1, 3), (2, 3), and (4, 5), for the pairs of entries (2, 1), (3, 1), and (5, 4).
Sometimes an inversion is defined as the pair of values whose order is reversed; this makes no difference for the "number" of inversions, and this pair (reversed) is also an inversion in the above sense for the inverse permutation "σ"−1. The number of inversions is an important measure for the degree to which the entries of a permutation are out of order; it is the same for "σ" and for "σ"−1. To bring a permutation with "k" inversions into order (that is, transform it into the identity permutation), by successively applying (right-multiplication by) adjacent transpositions, is always possible and requires a sequence of "k" such operations. Moreover, any reasonable choice for the adjacent transpositions will work: it suffices to choose at each step a transposition of "i" and where "i" is a descent of the permutation as modified so far (so that the transposition will remove this particular descent, although it might create other descents). This is so because applying such a transposition reduces the number of inversions by 1; as long as this number is not zero, the permutation is not the identity, so it has at least one descent. Bubble sort and insertion sort can be interpreted as particular instances of this procedure to put a sequence into order. Incidentally this procedure proves that any permutation "σ" can be written as a product of adjacent transpositions; for this one may simply reverse any sequence of such transpositions that transforms "σ" into the identity. In fact, by enumerating all sequences of adjacent transpositions that would transform "σ" into the identity, one obtains (after reversal) a "complete" list of all expressions of minimal length writing "σ" as a product of adjacent transpositions.
The number of permutations of "n" with "k" inversions is expressed by a Mahonian number, it is the coefficient of "X""k" in the expansion of the product
which is also known (with "q" substituted for "X") as the q-factorial ["n"]"q"! . The expansion of the product appears in Necklace (combinatorics).
Permutations in computing.
Numbering permutations.
One way to represent permutations of "n" is by an integer "N" with 0 ≤ "N" < "n"!, provided convenient methods are given to convert between the number and the representation of a permutation as an ordered arrangement (sequence). This gives the most compact representation of arbitrary permutations, and in computing is particularly attractive when "n" is small enough that "N" can be held in a machine word; for 32-bit words this means "n" ≤ 12, and for 64-bit words this means "n" ≤ 20. The conversion can be done via the intermediate form of a sequence of numbers "d""n", "d""n"−1, ..., "d"2, "d"1, where "d""i" is a non-negative integer less than "i" (one may omit "d"1, as it is always 0, but its presence makes the subsequent conversion to a permutation easier to describe). The first step then is to simply express "N" in the "factorial number system", which is just a particular mixed radix representation, where for numbers up to "n"! the bases for successive digits are "n", , ..., 2, 1. The second step interprets this sequence as a Lehmer code or (almost equivalently) as an inversion table.
In the Lehmer code for a permutation "σ", the number "d""n" represents the choice made for the first term "σ"1, the number "d""n"−1 represents the choice made for the second term
"σ"2 among the remaining elements of the set, and so forth. More precisely, each "d""n"+1−"i" gives the number of "remaining" elements strictly less than the term "σ""i". Since those remaining elements are bound to turn up as some later term "σ""j", the digit "d""n"+1−"i" counts the "inversions" ("i","j") involving "i" as smaller index (the number of values "j" for which "i" < "j" and "σ""i" > "σ""j"). The inversion table for "σ" is quite similar, but here "d""n"+1−"k" counts the number of inversions ("i","j") where "k" = "σ""j" occurs as the smaller of the two values appearing in inverted order. Both encodings can be visualized by an "n" by "n" Rothe diagram (named after Heinrich August Rothe) in which dots at ("i","σ""i") mark the entries of the permutation, and a cross at ("i","σ""j") marks the inversion ("i","j"); by the definition of inversions a cross appears in any square that comes both before the dot ("j","σ""j") in its column, and before the dot ("i","σ""i") in its row. The Lehmer code lists the numbers of crosses in successive rows, while the inversion table lists the numbers of crosses in successive columns; it is just the Lehmer code for the inverse permutation, and vice versa.
To effectively convert a Lehmer code "d""n", "d""n"−1, ..., "d"2, "d"1 into a permutation of an ordered set "S", one can start with a list of the elements of "S" in increasing order, and for "i" increasing from 1 to "n" set "σ""i" to the element in the list that is preceded by "d""n"+1−"i" other ones, and remove that element from the list. To convert an inversion table "d""n", "d""n"−1, ..., "d"2, "d"1 into the corresponding permutation, one can traverse the numbers from "d"1 to "d""n" while inserting the elements of "S" from largest to smallest into an initially empty sequence; at the step using the number "d" from the inversion table, the element from "S" inserted into the sequence at the point where it is preceded by "d" elements already present. Alternatively one could process the numbers from the inversion table and the elements of "S" both in the opposite order, starting with a row of "n" empty slots, and at each step place the element from "S" into the empty slot that is preceded by "d" other empty slots.
Converting successive natural numbers to the factorial number system produces those sequences in lexicographic order (as is the case with any mixed radix number system), and further converting them to permutations preserves the lexicographic ordering, provided the Lehmer code interpretation is used (using inversion tables, one gets a different ordering, where one starts by comparing permutations by the "place" of their entries 1 rather than by the value of their first entries). The sum of the numbers in the factorial number system representation gives the number of inversions of the permutation, and the parity of that sum gives the signature of the permutation. Moreover, the positions of the zeroes in the inversion table give the values of left-to-right maxima of the permutation (in the example 6, 8, 9) while the positions of the zeroes in the Lehmer code are the positions of the right-to-left minima (in the example positions the 4, 8, 9 of the values 1, 2, 5); this allows computing the distribution of such extrema among all permutations. A permutation with Lehmer code "d""n", "d""n"−1, ..., "d"2, "d"1 has an ascent if and only if .
Algorithms to generate permutations.
In computing it may be required to generate permutations of a given sequence of values. The methods best adapted to do this depend on whether one wants some randomly chosen permutations, or all permutations, and in the latter case if a specific ordering is required. Another question is whether possible equality among entries in the given sequence is to be taken into account; if so, one should only generate distinct multiset permutations of the sequence.
An obvious way to generate permutations of "n" is to generate values for the Lehmer code (possibly using the factorial number system representation of integers up to "n"!), and convert those into the corresponding permutations. However, the latter step, while straightforward, is hard to implement efficiently, because it requires "n" operations each of selection from a sequence and deletion from it, at an arbitrary position; of the obvious representations of the sequence as an array or a linked list, both require (for different reasons) about "n"2/4 operations to perform the conversion. With "n" likely to be rather small (especially if generation of all permutations is needed) that is not too much of a problem, but it turns out that both for random and for systematic generation there are simple alternatives that do considerably better. For this reason it does not seem useful, although certainly possible, to employ a special data structure that would allow performing the conversion from Lehmer code to permutation in "O"("n" log "n") time.
Random generation of permutations.
For generating random permutations of a given sequence of "n" values, it makes no difference whether one applies a randomly selected permutation of "n" to the sequence, or chooses a random element from the set of distinct (multiset) permutations of the sequence. This is because, even though in case of repeated values there can be many distinct permutations of "n" that result in the same permuted sequence, the number of such permutations is the same for each possible result. Unlike for systematic generation, which becomes unfeasible for large "n" due to the growth of the number "n"!, there is no reason to assume that "n" will be small for random generation.
The basic idea to generate a random permutation is to generate at random one of the "n"! sequences of integers "d"1,"d"2...,"d""n" satisfying (since "d"1 is always zero it may be omitted) and to convert it to a permutation through a bijective correspondence. For the latter correspondence one could interpret the (reverse) sequence as a Lehmer code, and this gives a generation method first published in 1938 by Ronald Fisher and Frank Yates.
While at the time computer implementation was not an issue, this method suffers from the difficulty sketched above to convert from Lehmer code to permutation efficiently. This can be remedied by using a different bijective correspondence: after using "d""i" to select an element among "i" remaining elements of the sequence (for decreasing values of "i"), rather than removing the element and compacting the sequence by shifting down further elements one place, one swaps the element with the final remaining element. Thus the elements remaining for selection form a consecutive range at each point in time, even though they may not occur in the same order as they did in the original sequence. The mapping from sequence of integers to permutations is somewhat complicated, but it can be seen to produce each permutation in exactly one way, by an immediate induction. When the selected element happens to be the final remaining element, the swap operation can be omitted. This does not occur sufficiently often to warrant testing for the condition, but the final element must be included among the candidates of the selection, to guarantee that all permutations can be generated.
The resulting algorithm for generating a random permutation of codice_1 can be described as follows in pseudocode:
 for "i" from "n" downto 2 do
 swap "a"["di"] and "a"["i" − 1]
This can be combined with the initialization of the array codice_2 as follows
 for "i" from 0 to "n"−1 do
 "a"["i"] ← "a"["d""i"+1]
 "a"["d""i"+1] ← "i"
If "d""i"+1 = "i", the first assignment will copy an uninitialized value, but the second will overwrite it with the correct value "i".
However, Fisher-Yates is not the fastest algorithm for generating a permutation, because Fisher-Yates is essentially a sequential algorithm and "divide and conquer" procedures can achieve the same result in parallel.
Generation in lexicographic order.
There are many ways to systematically generate all permutations of a given sequence.
One classic, simple, and flexible algorithm is based upon finding the next permutation in lexicographic ordering, if it exists. It can handle repeated values, for which case it generates each distinct multiset permutation once. Even for ordinary permutations it is significantly more efficient than generating values for the Lehmer code in lexicographic order (possibly using the factorial number system) and converting those to permutations. It begins by sorting the sequence in (weakly) increasing order (which gives its lexicographically minimal permutation), and then repeats advancing to the next permutation as long as one is found. The method goes back to Narayana Pandita in 14th century India, and has been rediscovered frequently.
The following algorithm generates the next permutation lexicographically after a given permutation. It changes the given permutation in-place.
For example, given the sequence [1, 2, 3, 4] (which is in increasing order), and given that the index is zero-based, the steps are as follows:
Following this algorithm, the next lexicographic permutation will be [1,3,2,4], and the 24th permutation will be [4,3,2,1] at which point "a"["k"] < "a"["k" + 1] does not exist, indicating that this is the last permutation.
This method uses about 3 comparisons and 1.5 swaps per permutation, amortized over the whole sequence, not counting the initial sort.
Generation with minimal changes.
An alternative to the above algorithm, the Steinhaus–Johnson–Trotter algorithm, generates an ordering on all the permutations of a given sequence with the property that any two consecutive permutations in its output differ by swapping two adjacent values. This ordering on the permutations was known to 17th-century English bell ringers, among whom it was known as "plain changes". One advantage of this method is that the small amount of change from one permutation to the next allows the method to be implemented in constant time per permutation. The same can also easily generate the subset of even permutations, again in constant time per permutation, by skipping every other output permutation.
An alternative to Steinhaus–Johnson–Trotter is Heap's algorithm, said by Robert Sedgewick in 1977 to be the fastest algorithm of generating permutations in applications.
The following figure shows the output of all three aforementioned algorithms for generating all permutations of length formula_129, and of six additional algorithms described in the literature.
Meandric permutations.
Meandric systems give rise to "meandric permutations", a special subset of "alternate permutations". An alternate permutation of the set {1, 2, ..., 2"n"} is a cyclic permutation (with no fixed points) such that the digits in the cyclic notation form alternate between odd and even integers. Meandric permutations are useful in the analysis of RNA secondary structure. Not all alternate permutations are meandric. A modification of Heap's algorithm has been used to generate all alternate permutations of order "n" (that is, of length 2"n") without generating all (2"n")! permutations. Generation of these alternate permutations is needed before they are analyzed to determine if they are meandric or not.
The algorithm is recursive. The following table exhibits a step in the procedure. In the previous step, all alternate permutations of length 5 have been generated. Three copies of each of these have a "6" added to the right end, and then a different transposition involving this last entry and a previous entry in an even position is applied (including the identity; that is, no transposition).
Applications.
Permutations are used in the interleaver component of the error detection and correction algorithms, such as turbo codes, for example 3GPP Long Term Evolution mobile telecommunication standard uses these ideas (see 3GPP technical specification 36.212).
Such applications raise the question of fast generation of permutations satisfying certain desirable properties. One of the methods is based on the permutation polynomials. Also as a base for optimal hashing in Unique Permutation Hashing.