Permutation

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In mathematics, the notion of permutation relates to the act of permuting, or rearranging, members of a set into a particular sequence or order (unlike combinations, which are selections that disregard order). For example, 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). As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters. The study of permutations of finite sets is a topic in the field of combinatorics.

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.

Permutations occur, in more or less prominent ways, in almost every area of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of sorting algorithms in computer science.

In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself (i.e., a map SS for which every element of S occurs exactly once as image value). This is related to the rearrangement of S in which each element s takes the place of the corresponding f(s). The collection of such permutations form a symmetric group. The key to its structure is the fact that the composition of two permutations (performing two given rearrangements in succession) defines a third rearrangement. Permutations may act on structured objects by rearranging their components, or by certain replacements (substitutions) of symbols.

In elementary combinatorics, the k-permutations, or partial permutations, are the sequences 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.

History[edit]

The rule to determine the number of permutations of n objects was known in Indian culture at least as early as 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.[1]

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.

Definition and usage[edit]

There are two common ways of regarding permutations.[2] They are completely equivalent and either form is readily converted to the other. Which form is preferable depends on the type of questions being asked about the permutations. Some disciplines use one form more predominantly than the other.

The first way to regard permutations of a set S (which can even be applied to infinite sets) is to define them as the bijections from S to itself. Thus, the permutations are being thought of as functions and so, can be composed with each other, forming groups of permutations. From this viewpoint, the elements of S have no special properties and are just being used as convenient names for the objects being moved around according to the bijection.[3]

In Cauchy's two-line notation,[4] one lists the elements of S in the first row, and for each one its image under the permutation below it in the second row. For instance, a particular permutation of the set {1,2,3,4,5} can be written as:

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

this means that σ satisfies σ(1)=2, σ(2)=5, σ(3)=4, σ(4)=3, and σ(5)=1. There is no special order that the elements of S appearing in the first row have to appear in. This permutation could also be written as:

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

However, if there is a "natural" order that the elements of S can be placed in, say x_1,x_2,\ldots,x_n, then under the assumption that the first row of the permutation,

\begin{pmatrix}
x_1 & x_2 & x_3 & \cdots & x_n \\
\sigma(x_1) &\sigma(x_2) & \sigma(x_3) & \cdots& \sigma(x_n)\end{pmatrix},

is in this natural order,[5] the first row need not be written. Thus, under this assumption, the permutation can also be written in one-line notation as (\sigma(x_1) \; \sigma(x_2) \; \sigma(x_3) \; \cdots \; \sigma(x_n)), which is the second common way of representing permutations.[6][7] That is, a permutation of the set S is an ordered arrangement (or listing, or linearly ordered arrangement) of the elements of S.[8] We shall use the term "sequence" in this article for an ordered arrangement although we only mean sequences in which the elements of S appear exactly once. The permutation \sigma above would then be given by (2 5 4 3 1) since the natural order (1 2 3 4 5) would be assumed. (It is typical to use commas to separate these entries only if some have two or more digits.) This form of representation is common in elementary combinatorics, computer science and those areas of combinatorics that are closely related to it. In many applications where the elements of S will be compared to each other, this is the preferred form of permutation representation (such applications require S to be a totally ordered set.[9])

There are n! permutations of a finite set S having n elements.

There is also a weaker meaning of the term "permutation" that is sometimes used in elementary combinatorics texts, designating those sequences in which no element occurs more than once, but without the requirement to use all 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 sequences of a fixed length k of elements taken from a given set of size n. These objects are also known as partial permutations or as sequences without repetition, terms that avoids confusion with the other, more common, meaning of "permutation". The number of such k-permutations of n is denoted variously by such symbols as n Pk, nPk, Pn,k, or P(n,k), and its value is given by the product[10]

n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)

which is 0 when k > n, and otherwise is equal to

\frac{n!}{(n-k)!}.

The product is well defined without the assumption that n is a non-negative integer and is of importance outside combinatorics as well; it is known as the Pochhammer symbol (n)k or as the k-th falling factorial power nk of n.


Permutations of multisets[edit]

Permutations of multisets

If M is a finite multiset, then a multiset permutation is a sequence of elements of M in which each element appears exactly as often as is its multiplicity in M. An anagram of a word having some repeated letters is an example of a multiset permutation.[11] If the multiplicities of the elements of M (taken in some order) are m_1, m_2, ..., m_l and their sum (i.e., the size of M) is n, then the number of multiset permutations of M is given by the multinomial coefficient,[12]

{n \choose m_1,m_2,\ldots,m_l} = \frac{n!}{m_1!\,m_2!\, \cdots\,m_l!}.

For example, the number of distinct anagrams of the word MISSISSIPPI is:[13]

\frac{11!}{1!4!4!2!}.

A k-permutation of a multiset M is a sequence of length k of elements of M in which each element appears at most its multiplicity in M times (an element's repetition number). Infinite repetition numbers are allowed in some applications.

Permutations in group theory[edit]

The set of all permutations of any given set S forms a group, with the composition of maps as the product operation and the identity function as the neutral element of the group. This is the symmetric group of S, denoted by Sym(S). Up to isomorphism, this symmetric group only depends on the cardinality of the set (called the degree of the group), so the nature of elements of S is irrelevant for the structure of the group. Symmetric groups have been studied mostly in the case of finite sets, so, confined to this case, one can assume without loss of generality that S = {1,2,...,n} for some natural number n. This is then the symmetric group of degree n, usually written as Sn.

Any subgroup of a symmetric group is called a permutation group. By Cayley's theorem any group is isomorphic to some permutation group, and every finite group to a subgroup of some finite symmetric group.

Cycle notation[edit]

Another notation for permutations called cycle notation focuses on the effect of successively applying the permutation. It expresses the permutation as a product of cycles corresponding to the orbits of the permutation; since distinct orbits are disjoint, this is referred to as "the decomposition into disjoint cycles" of the permutation. Due to the likely possibility of confusion, cycle notation is not used in conjunction with one-line notation (sequences) for permutations. It works as follows: starting from some element x of S, one writes the sequence (x σ(x) σ(σ(x)) ...) of successive images under σ, until the image would be x, at which point one instead closes the parenthesis. The set of values written down forms the orbit (under σ) of x, and the parenthesized expression gives the corresponding cycle of σ. One then continues choosing an element y of S that is not in the orbit already written down, and writes down the corresponding cycle, and so on until all elements of S belong to some cycle written down. 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 for instance

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

Each cycle (x1 x2 ... xk) of σ, denotes a permutation in its own right, namely the one that takes the same values as σ on this orbit (so it maps xi to xi+1 for i < k, and xk to x1), while mapping all other elements of S to themselves. The size k of the orbit is called the length of the cycle, and a cycle of length k is called a k-cycle. Any 1-cycle is the identity permutation and so, they are all the same permutation.[14][15] Distinct orbits of σ are by definition disjoint, so the corresponding cycles commute (as elements of the permutation group), and σ is the product of its cycles (taken in any order).[16] Therefore the concatenation of cycles in the cycle notation is interpreted as denoting composition (product) of permutations, and writing a permutation as a product of its cycles is called a decomposition into cycles of the permutation. This decomposition is essentially unique: apart from the reordering the cycles in the product, there are no other ways to write σ as a product of cycles. The cycle notation is less unique, since each individual cycle can be written in different ways, as in the example above where (5 1 2) denotes the same cycle as (1 2 5) or (2 5 1) (though note that (5 2 1) denotes a different cycle). In writing a permutation as a product of its cycles it is typical, but not required, to suppress the writing of 1-cycles when no confusion can arise.[17]

An orbit of size 1 (more precisely, the element of S in a 1-cycle) is called a fixed point of the permutation. A permutation that has no fixed point is called a derangement. Cycles of length two are called transpositions; such permutations merely exchange the place of two elements. Since the orbits of a permutation partition the set S, for a finite set of size n, the lengths of the cycles of a permutation σ form a partition of n called the cycle type of σ. 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 β = (1 2 5)(3 4)(6 8)(7), is (3,2,2,1) which is sometimes written in a more compact form as (11,22,31).

Permutation groups have more structure than abstract groups, different realizations of a group as a permutation group need not be equivalent for this additional structure. For instance S3 is naturally a permutation group, in which any transposition has cycle type (2,1), but the proof of Cayley's theorem realizes S3 as a subgroup of S6 (namely the permutations of the 6 elements of S3 itself), in which permutation group transpositions have cycle type (2,2,2). So in spite of Cayley's theorem, the study of permutation groups differs from the study of abstract groups.

Product and inverse[edit]

The product of two permutations is defined as their composition as functions, in other words σ·π is the function that maps any element x of the set to σ(π(x)). Note that the rightmost permutation is applied to the argument first, [18] because of the way function application is written. Some authors prefer the leftmost factor acting first, [19] [20] [21] but to that end permutations must be written to the right of their argument, for instance 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.

Since the composition of two bijections always gives another bijection, the product of two permutations is again a permutation. Since function composition is associative, so is the product operation on permutations: (σ·πρ = σ·(π·ρ). 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 multiplication.

The identity permutation, which maps every element of the set to itself, is the neutral element for this product. In two-line notation, the identity is

\begin{pmatrix}1 & 2 & 3 & \cdots & n \\ 1 & 2 & 3 & \cdots & n\end{pmatrix}.

Since bijections have inverses, so do permutations, and the inverse σ−1 of σ is again a permutation. Explicitly, whenever σ(x)=y one also has σ−1(y)=x. In two-line notation the inverse can be obtained by interchanging the two lines (and sorting the columns if one wishes the first line to be in a given order). For instance

\begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix}^{-1}
      =\begin{pmatrix}2 & 5 & 4 & 3 & 1\\ 1 & 2 & 3 & 4 & 5 \end{pmatrix}
      =\begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 5 & 1 & 4 & 3 & 2\end{pmatrix}.

In cycle notation one can reverse the order of the elements in each cycle to obtain a cycle notation for its inverse.

Having an associative product, a neutral element, and inverses for all its elements, makes the set of all permutations of S into a group, called the symmetric group of S.

Properties[edit]

Every permutation of a finite set can be expressed as the product of transpositions.[22] Moreover, although many such expressions for a given permutation may exist, there can never be among them both expressions with an even number and expressions with an odd number of transpositions. All permutations are then classified as even or odd, according to the parity of the transpositions in any such expression.

Composition of permutations corresponds to multiplication of permutation matrices.

Multiplying permutations written in cycle notation follows no easily described pattern, and the cycles of the product can be entirely different from those of the permutations being composed. However the cycle structure is preserved in the special case of conjugating a permutation σ by another permutation π, which means forming the product π·σ·π−1. Here the cycle notation of the result can be obtained by taking the cycle notation for σ and applying π to all the entries in it.[23]

Matrix representation[edit]

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 Mi,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.
Here (file) 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.

Permutation of components of a sequence[edit]

As with any group, one can consider actions of a symmetric group on a set, and there are many ways in which such an action can be defined. For the symmetric group of {1, 2, ..., n} there is one particularly natural action, namely the action by permutation on the set Xn of sequences of n symbols taken from some set X. Like for the matrix representation, there are two natural ways in which the result of permuting a sequence (x1,x2,...,xn) by σ can be defined, but only one is compatible with the multiplication of permutations (so as to give a left action of the symmetric group on Xn); with the multiplication rule used in this article this is the one given by

\sigma\cdot(x_1,\ldots,x_n) = (x_{\sigma^{-1}(1)},\ldots,x_{\sigma^{-1}(n)}).

This means that each component xi ends up at position σ(i) in the sequence permuted by σ.

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.

Here are a number of properties that are directly related to the total ordering of S.

Ascents, descents and runs[edit]

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 = σ1σ2...σn, then i is an ascent if σi < σi+1.

For example, the permutation 3452167 has ascents (at positions) 1,2,5,6.

Similarly, a descent is a position i < n with σi > σi+1, so every i with 1\leq i<n either is an ascent or is a descent of σ.

The number of permutations of n with k ascents is the Eulerian number \textstyle\left\langle{n\atop k}\right\rangle; this is also the number of permutations of n with k descents.[24]

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. Hence, the number of permutations of n with k ascending runs is the same as the number \textstyle\left\langle{n\atop k-1}\right\rangle of permutations with k − 1 descents.[25]

Inversions[edit]

An inversion of a permutation σ is a pair (i,j) of positions where the entries of a permutation are in the opposite order: i<j and \sigma_i>\sigma_j.[26] So a descent is just an inversion at two adjacent positions. For example, the permutation σ = 23154 has three inversions: (1,3), (2,3), (4,5), for the pairs of entries (2,1), (3,1), (5,4).

Sometimes an inversion is defined as the pair of values (σi,σj) itself 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 (i.e., 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 i + 1 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; also note that 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,[27] it is the coefficient of Xk in the expansion of the product

\prod_{m=1}^n\sum_{i=0}^{m-1}X^i=1(1+X)(1+X+X^2)\cdots(1+X+X^2+\cdots+X^{n-1}),

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[edit]

Numbering permutations[edit]

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 usual representation of a permutation as a 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 dn, dn−1, ..., d2, d1, where di is a non-negative integer less than i (one may omit d1, as it is always 0, but its presence makes the subsequent conversion to a permutation easier to describe). The first step then is simply expression of 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, n − 1, ..., 2, 1. The second step interprets this sequence as a Lehmer code or (almost equivalently) as an inversion table.

Rothe diagram for \sigma=(6,3,8,1,4,9,7,2,5)
i  \ σi 1 2 3 4 5 6 7 8 9 Lehmer code
1 × × × × × d9 = 5
2 × × d8 = 2
3 × × × × × d7 = 5
4 d6 = 0
5 × d5 = 1
6 × × × d4 = 3
7 × × d3 = 2
8 d2 = 0
9 d1 = 0
inversion table 3 6 1 2 4 0 2 0 0

In the Lehmer code for a permutation σ, the number dn represents the choice made for the first term σ1, the number dn−1 represents the choice made for the second term σ2 among the remaining n − 1 elements of the set, and so forth. More precisely, each dn+1−i gives the number of remaining elements strictly less than the term σi. Since those remaining elements are bound to turn up as some later term σj, the digit dn+1−i counts the inversions (i,j) involving i as smaller index (the number of values j for which i < j and σi > σj). The inversion table for σ is quite similar, but here dn+1−k counts the number of inversions (i,j) where k = σj occurs as the smaller of the two values appearing in inverted order.[28] Both encodings can be visualized by an n by n Rothe diagram[29] (named after Heinrich August Rothe) in which dots at (i,σi) mark the entries of the permutation, and a cross at (i,σj) marks the inversion (i,j); by the definition of inversions a cross appears in any square that comes both before the dot (j,σj) in its column, and before the dot (i,σi) 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 dn, dn−1, ..., d2, d1 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 σi to the element in the list that is preceded by dn+1−i other ones, and remove that element from the list. To convert an inversion table dn, dn−1, ..., d2, d1 into the corresponding permutation, one can traverse the numbers from d1 to dn 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 dn, dn−1, ..., d2, d1 has an ascent ni if and only if didi+1.

Algorithms to generate permutations[edit]

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 n2/4 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[edit]

For generating random permutations of a given sequence of n values, it makes no difference whether one means apply a randomly selected permutation of n to the sequence, or choose 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 d1,d2,...,dn satisfying 0 ≤ di < i (since d1 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 A. Fisher and Frank Yates.[30] 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 di 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 a[0], a[1], ..., a[n − 1] can be described as follows in pseudocode:

for i from n downto 2
do   di ← random element of { 0, ..., i − 1 }
swap a[di] and a[i − 1]

This can be combined with the initialization of the array a[i] = i as follows:

for i from 0 to n−1
do   di+1 ← random element of { 0, ..., i }
a[i] ← a[di+1]
a[di+1] ← i

If di+1 = i, the first assignment will copy an uninitialized value, but the second will overwrite it with the correct value i.

Generation in lexicographic order[edit]

There are many ways to systematically generate all permutations of a given sequence.[31] One classical algorithm, which is both simple and flexible, is based on finding the next permutation in lexicographic ordering, if it exists. It can handle repeated values, for which case it generates the distinct multiset permutations each 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. To use it, one starts 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 frequently rediscovered ever since.[32]

The following algorithm generates the next permutation lexicographically after a given permutation. It changes the given permutation in-place.

  1. Find the largest index k such that a[k] < a[k + 1]. If no such index exists, the permutation is the last permutation.
  2. Find the largest index l such that a[k] < a[l].
  3. Swap the value of a[k] with that of a[l].
  4. Reverse the sequence from a[k + 1] up to and including the final element a[n].

For example, given the sequence [1, 2, 3, 4] which starts in a weakly increasing order, and given that the index is zero-based, the steps are as follows:

  1. Index k = 2, because 3 is placed at an index that satisfies condition of being the largest index that is still less than a[k + 1] which is 4.
  2. Index l = 3, because 4 is the only value in the sequence that is greater than 3 in order to satisfy the condition a[k] < a[l].
  3. The values of a[2] and a[3] are swapped to form the new sequence [1,2,4,3].
  4. The sequence after k-index a[2] to the final element is reversed. Because only one value lies after this index (the 3), the sequence remains unchanged in this instance. Thus the lexicographic successor of the initial state is permuted: [1,2,4,3].

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.

Generation with minimal changes[edit]

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.[32] An old paper[33] proposed a very efficient algorithm called Heap's permutation generation algorithm. An article[31] authored by Robert Sedgewick said that this algorithm is the fastest algorithm of generating permutations in applications.

Meandric permutations[edit]

Meandric systems give rise to meandric permutations, a special subset of alternate permutations. An alternate permutation of the set {1,2,...,2n} 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 2n) without generating all (2n)! permutations.[34] 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, i.e., no transposition).

Previous sets Transposition of digits Alternate permutations
1-2-3-4-5-6 1-2-3-4-5-6
1-2-3-4-5-6 4,6 1-2-3-6-5-4
1-2-3-4-5-6 2,6 1-6-3-4-5-2
1-2-5-4-3-6 1-2-5-4-3-6
1-2-5-4-3-6 4,6 1-2-5-6-3-4
1-2-5-4-3-6 2,6 1-6-5-4-3-2
1-4-3-2-5-6 1-4-3-2-5-6
1-4-3-2-5-6 2,6 1-4-3-6-5-2
1-4-3-2-5-6 4,6 1-6-3-2-5-4
1-4-5-2-3-6 1-4-5-2-3-6
1-4-5-2-3-6 2,6 1-4-5-6-3-2
1-4-5-2-3-6 4,6 1-6-5-2-3-4

Software implementations[edit]

Calculator functions[edit]

Many scientific calculators and computing software have a built-in function for calculating the number of k-permutations of n.

  • Casio and TI calculators: nPr
  • HP calculators: PERM[35]
  • Mathematica: FallingFactorial

Spreadsheet functions[edit]

Most spreadsheet software also provides a built-in function for calculating the number of k-permutations of n, called PERMUT in many popular spreadsheets.

Applications[edit]

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 [36]). Such applications raise the question of fast generation of permutations satisfying certain desirable properties. One of the methods is based on the permutation polynomials.

See also[edit]

Notes[edit]

  1. ^ N. L. Biggs, The roots of combinatorics, Historia Math. 6 (1979) 109−136
  2. ^ Cameron 1994, p. 28
  3. ^ In older usage the term "permutation on n letters" was used to convey the lack of special properties of the objects being permuted.
  4. ^ Wussing, Hans (2007), The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, Courier Dover Publications, p. 94, ISBN 9780486458687, "Cauchy used his permutation notation—in which the arrangements are written one below the other and both are enclosed in parentheses—for the first time in 1815." 
  5. ^ Explicit mention of this natural order is often suppressed. When the set is {1,2,...,n}, the natural order is (1 2 ... n) and this is almost never mentioned. If the set consists of letters, the natural order is lexicographical. For other sets, a natural order needs to be made explicit.
  6. ^ Bogart 1990, p. 17
  7. ^ Gerstein 1987, p. 217
  8. ^ Cameron (1994) uses the terms "active" and "passive" to distinguish between the bijective form (active) and the ordered arrangement form (passive). In a footnote (p. 29, footnote 3) he mentions that 19th century mathematicians would more usually use the term "permutation" for the passive form and "substitution" for the active form.
  9. ^ As the set of natural numbers {1,2,...,n} is a totally ordered set under the usual "<" ordering, it is the most convenient set to use in these applications.
  10. ^ Charalambides, Ch A. (2002). Enumerative Combinatorics. CRC Press. p. 42. ISBN 978-1-58488-290-9. 
  11. ^ The natural order in this example is the order of the letters in the original word.
  12. ^ Brualdi 2010, p. 46, Theorem 2.4.2
  13. ^ Brualdi 2010, p. 47
  14. ^ Rotman 2002, p. 41
  15. ^ Bogart 1990, p. 487
  16. ^ Rotman 2002, p. 43
  17. ^ Hall 1959, p. 54
  18. ^ Biggs, Norman L.; White, A. T. (1979). Permutation groups and combinatorial structures. Cambridge University Press. ISBN 0-521-22287-7. 
  19. ^ Dixon, John D.; Mortimer, Brian (1996). Permutation Groups. Springer. ISBN 0-387-94599-7. 
  20. ^ Cameron, Peter J. (1999). Permutation groups. Cambridge University Press. ISBN 0-521-65302-9. 
  21. ^ Jerrum, M. (1986). "A compact representation of permutation groups". J. Algor. 7 (1): 60–78. 
  22. ^ Hall 1959, p. 60
  23. ^ Humphreys (1996), p. 84
  24. ^ Combinatorics of Permutations, ISBN 1-58488-434-7, M. Bóna, 2004, p. 3
  25. ^ Combinatorics of Permutations, ISBN 1-58488-434-7, M. Bóna, 2004, p. 4f
  26. ^ Combinatorics of Permutations, ISBN 1-58488-434-7, M. Bóna, 2004, p. 43
  27. ^ Combinatorics of Permutations, ISBN 1-58488-434-7, M. Bóna, 2004, p. 43ff
  28. ^ a b D. E. Knuth, The Art of Computer Programming, Vol 3, Sorting and Searching, Addison–Wesley (1973), p. 12. This book mentions the Lehmer code (without using that name) as a variant C1,...,Cn of inversion tables in exercise 5.1.1−7 (p. 19), together with two other variants.
  29. ^ H. A. Rothe, Sammlung combinatorisch-analytischer Abhandlungen 2 (Leipzig, 1800), 263−305. Cited in,[28] p. 14.
  30. ^ Fisher, R.A.; Yates, F. (1948) [1938]. Statistical tables for biological, agricultural and medical research (3rd ed.). London: Oliver & Boyd. pp. 26–27. OCLC 14222135. 
  31. ^ a b Sedgewick, R (1977). "Permutation generation methods". Computing Surveys 9: 137–164. 
  32. ^ a b Knuth, D. E. (2005). "Generating All Tuples and Permutations". The Art of Computer Programming. 4, Fascicle 2. Addison-Wesley. pp. 1–26. ISBN 0-201-85393-0. 
  33. ^ Heap, B. R. (1963). "Permutations by Interchanges". The Computer Journal 6 (3): 293–4. 
  34. ^ Alexiou, A.; Psiha, P.; Vlamos (2011). "Combinatorial permutation based algorithm for representation of closed RNA secondary structures". Bioinformation 7(2): 91–95. 
  35. ^ http://h20331.www2.hp.com/Hpsub/downloads/50gProbability-Rearranging_items.pdf
  36. ^ 3GPP TS 36.212

References[edit]

  • Bogart, Kenneth P. (1990), Introductory Combinatorics (2nd ed.), Harcourt Brace Jovanovich, ISBN 0-15-541576-X 
  • Bóna, Miklós (2004), Combinatorics of Permutations, Chapman Hall-CRC, ISBN 1-58488-434-7 
  • Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Prentice-Hall, ISBN 978-0-13-602040-0 
  • Cameron, Peter J. (1994), Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, ISBN 0-521-45761-0 
  • Gerstein, Larry J. (1987), Discrete Mathematics and Algebraic Structures, W.H. Freeman and Co., ISBN 0-7167-1804-9 
  • Donald Knuth. The Art of Computer Programming, Volume 4: Generating All Tuples and Permutations, Fascicle 2, first printing. Addison–Wesley, 2005. ISBN 0-201-85393-0.
  • Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Second Edition. Addison–Wesley, 1998. ISBN 0-201-89685-0. Section 5.1: Combinatorial Properties of Permutations, pp. 11–72.
  • Hall, Jr., Marshall (1959), The Theory of Groups, MacMillan 
  • Humphreys, J. F. (1996), A course in group theory, Oxford University Press, ISBN 978-0-19-853459-4 
  • Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice-Hall, ISBN 0-13-087868-5 

External links[edit]