TheInfoList

OR: In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a permutation of a set is, loosely speaking, an arrangement of its members into a
sequence In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
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
combination In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
s, which are selections of some members of a set regardless of order. For example, written as
tuple In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
s, there are six permutations of the set , 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.
Anagram An anagram is a word or phrase formed by rearranging the letters of a different word or phrase, typically using all the original letters exactly once. For example, the word ''anagram'' itself can be rearranged into ''nag a ram'', also the word ...
s 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 set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
s is an important topic in the fields of
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other ar ...
and
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. Permutations are used in almost every branch of mathematics, and in many other fields of science. In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, they are used for analyzing sorting algorithms; in
quantum physics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
, for describing states of particles; and in
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of Cell (biology), cells that proce ...
, for describing RNA sequences. The number of permutations of distinct objects is
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
, 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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
from to itself. That is, it is a function from to for which every element occurs exactly once as an
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 $\alpha$ defined as : $\alpha\left(1\right) = 3, \quad \alpha\left(2\right) = 1, \quad \alpha\left(3\right) = 2$. 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 -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 , can be seen as a compound polygon, compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram (Greek language, Greek) or sexagram (Latin l ...
were used in China in the
I Ching The ''I Ching'' or ''Yi Jing'' (, ), usually translated ''Book of Changes'' or ''Classic of Changes'', is an ancient Chinese divination text that is among the oldest of the Chinese classics. Originally a divination manual in the Western Z ...
(
Pinyin Hanyu Pinyin (), often shortened to just pinyin, is the official romanization system for Standard Chinese, Standard Mandarin Chinese in China, and to some extent, in Singapore and Malaysia. It is often used to teach Mandarin, normally writte ...
: 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 Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: * Twelvefold way Explained separately in a more accessible way: * Combination * Permutation In mathematics, a permutati ...
, to list all possible
Arabic Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
words with and without vowels. 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.
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 In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
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 $3! = 1 \cdot 2 \cdot 3 = 6$. 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 $nPk$ (which reads "n permute k"), and is equal to the number $n \left(n-1\right) \cdots \left(n - k + 1\right)$ (also written as

# Definition

In mathematics texts it is customary to denote permutations using lowercase Greek letters. Commonly, either $\alpha$ and $\beta$, or $\sigma, \tau$ and $\pi$ 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 $S_n$, where the group operation is
function composition In mathematics, function composition is an operation that takes two function (mathematics), functions and , and produces a function such that . In this operation, the function is function application, applied to the result of applying the ...
. Thus for two permutations, $\pi$ and $\sigma$ in the group $S_n$, the four group axioms hold: # Closure: If $\pi$ and $\sigma$ are in $S_n$ then so is $\pi\sigma.$ #
Associativity In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
: For any three permutations $\pi, \sigma, \tau \in S_n$, $\left(\pi\sigma\right)\tau = \pi\left(\sigma\tau\right).$ # Identity: There is an identity permutation, denoted $\operatorname$ and defined by $\operatorname\left(x\right) = x$ for all $x \in S$. For any $\sigma \in S_n$, $\operatorname \sigma = \sigma \operatorname = \sigma.$ # Invertibility: For every permutation $\pi \in S_n$, there exists an inverse permutation $\pi^ \in S_n$, so that $\pi\pi^ = \pi^\pi = \operatorname.$ In general, composition of two permutations is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, that is, $\pi\sigma \neq \sigma\pi.$ 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. A permutation can be decomposed into one or more disjoint ''cycles'', that is, the
orbits In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or pos ...
, which are found by repeatedly tracing the application of the permutation on some elements. For example, the permutation $\sigma$ defined by $\sigma\left(7\right) = 7$ has a 1-cycle, $\left(\,7\,\right)$ while the permutation $\pi$ defined by $\pi\left(2\right) = 3$ and $\pi\left(3\right) = 2$ has a 2-cycle $\left(\,2\,3\,\right)$ (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 $\left(\,x\,\right)$ 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 Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
'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'' = can be written as : $\sigma = \begin 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1 \end;$ 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: : $\sigma = \begin 3 & 2 & 5 & 1 & 4 \\ 4 & 5 & 1 & 2 & 3 \end,$ or : $\sigma = \begin 5 & 1 & 4 & 3 & 2 \\ 1 & 2 & 3 & 4 & 5 \end.$

## One-line notation

If there is a "natural" order for the elements of ''S'', say $x_1, x_2, \ldots, x_n$, then one uses this for the first row of the two-line notation: : $\sigma = \begin x_1 & x_2 & x_3 & \cdots & x_n \\ \sigma\left(x_1\right) & \sigma\left(x_2\right) & \sigma\left(x_3\right) & \cdots & \sigma\left(x_n\right) \end.$ Under this assumption, one may omit the first row and write the permutation in ''one-line notation'' as : $\left(\sigma\left(x_1\right) \; \sigma\left(x_2\right) \; \sigma\left(x_3\right) \; \cdots \; \sigma\left(x_n\right)\right)$, that is, as an ordered arrangement of the elements 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 A word is a basic element of language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, ...
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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other ar ...
and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
. 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 $\sigma$ in cycle notation, one proceeds as follows: # Write an opening bracket then select an arbitrary element ''x'' of $S$ and write it down: $\left(\,x$ # Then trace the orbit of ''x''; that is, write down its values under successive applications of $\sigma$: $\left(\,x\,\sigma\left(x\right)\,\sigma\left(\sigma\left(x\right)\right)\,\ldots$ # Repeat until the value returns to ''x'' and write down a closing parenthesis rather than ''x'': $\left(\,x\,\sigma\left(x\right)\,\sigma\left(\sigma\left(x\right)\right)\,\ldots\,\right)$ # Now continue with an element ''y'' of ''S'', not yet written down, and proceed in the same way: $\left(\,x\,\sigma\left(x\right)\,\sigma\left(\sigma\left(x\right)\right)\,\ldots\,\right)\left(\,y\,\ldots\,\right)$ # Repeat until all elements of ''S'' are written in cycles. So the permutation (in one-line notation) could be written as in cycle notation. While permutations in general do not commute, disjoint cycles do; for example, $(\,1 \, 2 \, 5\,) (\,3\,4\,) = (\,3 \, 4\,) (\,1 \, 2 \, 5\,).$ In addition, each cycle can be written in different ways, by choosing different starting points; for example, $(\,1 \, 2 \, 5\,) (\,3\,4\,) = (\,5 \, 1 \, 2\,)(\,3 \, 4\,) = (\,2 \, 5 \, 1\,)(\,4 \, 3\,).$ 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 $\sigma\left(x\right) = x$. 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 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

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. 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. $\sigma\cdot \pi$ is the function that maps any element ''x'' of the set to $\sigma\left(\pi\left(x\right)\right)$. The rightmost permutation is applied to the argument first, because of the way the function application is written. Since
function composition In mathematics, function composition is an operation that takes two function (mathematics), functions and , and produces a function such that . In this operation, the function is function application, applied to the result of applying the ...
is associative, so is the composition operation on permutations: $\left(\sigma\pi\right)\tau = \sigma\left(\pi\tau\right)$. 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 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 $k$-permutations of $n$ is denoted variously by such symbols as $P^n_k$, $_nP_k$, $^nP_k$, $P_$, or $P\left(n,k\right)$, and its value is given by the product : $P\left(n,k\right) = \underbrace_$, which is 0 when , and otherwise is equal to : $\frac.$ 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 $\left(n\right)_k$ or as the $k$-th falling factorial power $n^$ of $n$. This usage of the term ''permutation'' is closely related to the term ''
combination In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
''. 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: : $C\left(n,k\right) = \frac = \frac = \frac.$ These numbers are also known as
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the ter ...
s and are denoted by $\binom$.

## Permutations with repetition

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 $n^k.$ 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

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 An anagram is a word or phrase formed by rearranging the letters of a different word or phrase, typically using all the original letters exactly once. For example, the word ''anagram'' itself can be rearranged into ''nag a ram'', also the word ...
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 $m_1$, $m_2$, ..., $m_l$ 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, :$= \frac = \frac.$ For example, the number of distinct anagrams of the word MISSISSIPPI is: :$\frac = 34650$. 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
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           4

The 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          2
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 .

## Cycle type

The cycles (including the fixed points) of a permutation $\sigma$ of a set with elements partition that set; so the lengths of these cycles form an integer partition of , which is called the cycle type (or sometimes cycle structure or cycle shape) of $\sigma$. There is a "1" in the cycle type for every fixed point of $\sigma$, a "2" for every transposition, and so on. The cycle type of $\beta = \left(1\,2\,5\,\right)\left(\,3\,4\,\right)\left(6\,8\,\right)\left(\,7\,\right)$ is $\left(3, 2, 2, 1\right).$ This may also be written in a more compact form as . More precisely, the general form 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 type is preserved in the special case of conjugating a permutation $\sigma$ by another permutation $\pi$, which means forming the product $\pi\sigma\pi^$. Here, $\pi\sigma\pi^$ is the ''conjugate'' of $\sigma$ by $\pi$ and its cycle notation can be obtained by taking the cycle notation for $\sigma$ and applying $\pi$ to all the entries in it. It follows that two permutations are conjugate exactly when they have the same cycle type.

## Permutation order

The order of a permutation $\sigma$ is the smallest positive integer ''m'' so that $\sigma^m = \mathrm$. It is the
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...
of its cycles lengths. For example, the order of $\left(\,1\,3\,2\right)\left(\,4\,5\,\right)$ is $2\cdot3 = 6$.

## 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 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 $\operatorname\sigma$, to each permutation. $\operatorname\sigma = +1$ if $\sigma$ is even and $\operatorname\sigma = -1$ if $\sigma$ is odd. Then for two permutations $\sigma$ and $\pi$ : $\operatorname\left(\sigma\pi\right) = \operatorname\sigma\cdot\operatorname\pi.$ It follows that $\operatorname\left\left(\sigma\sigma^\right\right) = +1.$

## Matrix representation

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 . One natural approach is to associate to the permutation ''σ'' the matrix $M_$ 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, $M_\sigma M_\pi = M_$ for all permutations ''σ'' and ''π''. Second, if $_i$ represents the standard $n \times 1$
column vector In linear algebra, a column vector with m elements is an m \times 1 Matrix_(mathematics), matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times ...
(the vector with ''i''th entry equal to 1 and all other entries equal to 0), then $M_\sigma _i = _$. For example, with this convention, the matrix associated to the permutation $\sigma\left(1,2,3\right)=\left(2,1,3\right)$ is $\begin 0&1&0\\1&0&0\\0&0&1\end$ and the matrix associated to the permutation $\pi\left(1,2,3\right)=\left(2,3,1\right)$ is $\begin 0&0&1\\1&0&0\\0&1&0\end$. Then the composition of permutations is $\left(\sigma\circ\pi\right)\left(1,2,3\right)=\sigma\left(2,3,1\right)=\left(1,3,2\right)$, and the corresponding matrix product is $M_ M_ = \begin 0&1&0\\1&0&0\\0&0&1\end\begin 0&0&1\\1&0&0\\0&1&0\end = \begin 1&0&0\\0&0&1\\0&1&0\end = M_.$ It is also common in the literature to find the inverse convention, where a permutation ''σ'' is associated to the matrix $P_ = \left(M_\right)^ = \left(M_\right)^$ 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, $P_\sigma P_ = P_$ for all permutations ''σ'' and ''π''. In this correspondence, permutation matrices act by permuting indices of standard $1 \times n$ row vectors $\left(_i\right)^T$: one has $\left(_i\right)^T P_ = \left(_\right)^T$. The
Cayley table Named after the 19th century United Kingdom, British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an additio ...
on the right shows these matrices for permutations of 3 elements.

# 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
so that any two elements can be compared. The set 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 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.

## Foata's transition lemma

There is a relationship between the one-line notation and the canonical cycle notation. Consider the permutation $\left(\,2\,\right)\left(\,3\,1\,\right)$ in canonical cycle notation; if we simply remove the parentheses, we obtain the permutation $231$ 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 $f\left(p\right)=q$ be the parentheses-erasing transformation which returns $q$ in one-line notation when given $p$ in canonical cycle notation. As stated, $f$ operates by simply removing all parentheses. The operation of the inverse transformation, $f^\left(q\right)=p$, which returns $p$ in canonical cycle notation when given $q$ in one-line notation, is a bit less intuitive. Given the one-line notation $q = q_1q_2\cdots q_n$, the first cycle of $p$ in canonical cycle notation must start with $q_1$. As long as the subsequent elements are smaller than $q_1$, we are in the same cycle of $p$. The second cycle of $p$ starts at the smallest index $j$ such that $q_j > q_1$. In other words, $q_j$ 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 permutation $q=312548976$, 5 is the first element larger than the starting element 3, so the first cycle of $p$ must be $\left(\,3\,1\,2\,\right)$. Then 8 is the next element larger than 5, so the second cycle is $\left(\,5\,4\,\right)$. Since 9 is larger than 8, $\left(\,8\,\right)$ is a cycle by itself. Finally, 9 is larger than all the remaining elements to its right, so the last cycle is $\left(\,9\,7\,6\,\right)$. Concatenating these 4 cycles gives $p=\left(\,3\,1\,2\,\right)\left(\,5\,4\,\right)\left(\,8\,\right)\left(\,9\,7\,6\,\right)$ in canonical cycle notation. The following table shows both $q$ and $p$ for the six permutations of $123$. The bold side of each equality shows the permutation using its designated notation (one-line notation for $q$ and canonical cycle notation for $p$) while the non-bold side shows the same permutation in the other notation. Comparing the bold side of each column of the table shows the parenthesis removing/restoring operation of Foata's bijection, while comparing the same side of each column (for example, the LHS) shows which permutations are mapped to themselves by the bijection (first 3 rows) and which are not (last 3 rows). 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, $c\left(n, k\right)$. 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).

## Inversions

An '' inversion'' of a permutation ''σ'' is a pair of positions where the entries of a permutation are in the opposite order: $i < j$ and $\sigma_i > \sigma_j$. 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 Insertion sort is a simple sorting algorithm that builds the final sorted array (or list) one item at a time Comparison sort, by comparisons. It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or m ...
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 $\prod_^n\sum_^X^i = 1 \left(1 + X\right)\left(1 + X + X^2\right) \cdots \left(1 + X + X^2 + \cdots + X^\right),$ which is also known (with ''q'' substituted for ''X'') as the q-factorial /nowiki>''n''/nowiki>''q''! . The expansion of the product appears in Necklace (combinatorics). Let $\sigma \in S_n, i, j\in \$ such that

# Permutations in computing

## Numbering permutations

One way to represent permutations of ''n'' things 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 less than ''n''!, the bases (place values or multiplication factors) for successive digits are , , ..., 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 A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
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
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, 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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is pair ...
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 Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
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 ''a'' ''a'' ..., ''a'' 'n'' − 1/code> can be described as follows in pseudocode: for ''i'' from ''n'' downto 2 do ''di'' ← random element of 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 ''d''''i''+1 ← random element of ''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. # Find the largest index ''k'' such that . If no such index exists, the permutation is the last permutation. # Find the largest index ''l'' greater than ''k'' such that . # Swap the value of ''a'' 'k''with that of ''a'' 'l'' # Reverse the sequence from ''a'' 'k'' + 1up to and including the final element ''a'' 'n'' For example, given the sequence , 2, 3, 4(which is in increasing order), and given that the index is zero-based, the steps are as follows: # 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'' + 1which is 4. # 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'' # The values of ''a'' and ''a'' are swapped to form the new sequence , 2, 4, 3 # The sequence after ''k''-index ''a'' 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: , 2, 4, 3 Following this algorithm, the next lexicographic permutation will be , 3, 2, 4 and the 24th permutation will be , 3, 2, 1at which point ''a'' 'k''< ''a'' 'k'' + 1does 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 $n=4$, and of six additional algorithms described in the literature. # Lexicographic ordering; # Steinhaus–Johnson–Trotter algorithm; # Heap's algorithm; # Ehrlich's star-transposition algorithm: in each step, the first entry of the permutation is exchanged with a later entry; # Zaks' prefix reversal algorithm: in each step, a prefix of the current permutation is reversed to obtain the next permutation; # Sawada-Williams' algorithm: each permutation differs from the previous one either by a cyclic left-shift by one position, or an exchange of the first two entries; # Corbett's algorithm: each permutation differs from the previous one by a cyclic left-shift of some prefix by one position; # Single-track ordering: each column is a cyclic shift of the other columns; # Single-track Gray code: each column is a cyclic shift of the other columns, plus any two consecutive permutations differ only in one or two transpositions. Meandric permutations Meandric systems give rise to ''meandric permutations'', a special subset of ''alternate permutations''. An alternate permutation of the set 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 In information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by ... 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. See also * Alternating permutation * Convolution In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ... * Cyclic order * Even and odd permutations * Josephus permutation * Levi-Civita symbol * List of permutation topics * Major index * Permutation category * Permutation group * Permutation pattern * Permutation representation (symmetric group) * Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ... * Rencontres numbers * Sorting network * Substitution cipher * Superpattern * Superpermutation * Twelvefold way In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either ... * Weak order of permutations Notes References Bibliography * * * * * * * * * * * This book mentions the Lehmer code (without using that name) as a variant ''C''1,...,''C''''n'' of inversion tables in exercise 5.1.1–7 (p. 19), together with two other variants. * Fascicle 2, first printing. * * * * The publisher is given as "W.S." who may have been William Smith, possibly acting as agent for the Society of College Youths, to which society the "Dedicatory" is addressed. ''In quotations the original long "S" has been replaced by a modern short "s".'' * Further reading * * . The link is to a freely available retyped (LaTeX'ed) and revised version of the text originally published by Springer-Verlag. * . Section 5.1: Combinatorial Properties of Permutations, pp. 11–72. * * External links * {{Commons category, Permutations Arab inventions 
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