Two-line notation

Cauchy's two-line notation^[18] lists the elements of S in the first row, and the image of each element below it in the second row. For example, the permutation of S = {1, 2, 3, 4, 5, 6} given by the function

${\displaystyle \sigma (1)=2,\ \ \sigma (2)=6,\ \ \sigma (3)=5,\ \ \sigma (4)=4,\ \ \sigma (5)=3,\ \ \sigma (6)=1}$

can be written as

${\displaystyle \sigma ={\begin{pmatrix}1&2&3&4&5&6\\2&6&5&4&3&1\end{pmatrix}}.}$

The elements of S may appear in any order in the first row, so this permutation could also be written:

${\displaystyle \sigma ={\begin{pmatrix}2&3&4&5&6&1\\6&5&4&3&1&2\end{pmatrix}}={\begin{pmatrix}6&5&4&3&2&1\\1&3&4&5&6&2\end{pmatrix}}.}$

One-line notation

If there is a "natural" order for the elements of S,^{[lower-alpha 2]} say ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$, then one uses this for the first row of the two-line notation:

${\displaystyle \sigma ={\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\\sigma (x_{1})&\sigma (x_{2})&\sigma (x_{3})&\cdots &\sigma (x_{n})\end{pmatrix}}.}$

Under this assumption, one may omit the first row and write the permutation in one-line notation as

${\displaystyle \sigma =\sigma (x_{1})\;\sigma (x_{2})\;\sigma (x_{3})\;\cdots \;\sigma (x_{n})}$,

that is, as an ordered arrangement of the elements of S.^[19][20] Care must be taken to distinguish one-line notation from the cycle notation described below: a common usage is to omit parentheses or other enclosing marks for one-line notation, while using parentheses for cycle notation. The one-line notation is also called the word representation.^[21]

The example above would then be:

${\displaystyle \sigma ={\begin{pmatrix}1&2&3&4&5&6\\2&6&5&4&3&1\end{pmatrix}}=265431.}$

(It is typical to use commas to separate these entries only if some have two or more digits.)

This compact form is common in elementary combinatorics and computer science. It is especially useful in applications where the permutations are to be compared as larger or smaller using lexicographic order.

Cycle notation

Cycle notation describes the effect of repeatedly applying the permutation on the elements of the set S, with an orbit being called a cycle. The permutation is written as a list of cycles; since distinct cycles involve disjoint sets of elements, this is referred to as "decomposition into disjoint cycles".

To write down the permutation ${\displaystyle \sigma }$ in cycle notation, one proceeds as follows:

1. Write an opening bracket followed by an arbitrary element x of ${\displaystyle S}$: ${\displaystyle (\,x}$
2. Trace the orbit of x, writing down the values under successive applications of ${\displaystyle \sigma }$: ${\displaystyle (\,x,\sigma (x),\sigma (\sigma (x)),\ldots }$
3. Repeat until the value returns to x, and close the parenthesis without repeating x: ${\displaystyle (\,x\,\sigma (x)\,\sigma (\sigma (x))\,\ldots \,)}$
4. Continue with an element y of S which was not yet written, and repeat the above process: ${\displaystyle (\,x\,\sigma (x)\,\sigma (\sigma (x))\,\ldots \,)(\,y\,\ldots \,)}$
5. Repeat until all elements of S are written in cycles.

Also, it is common to omit 1-cycles, since these can be inferred: for any element x in S not appearing in any cycle, one implicitly assumes ${\displaystyle \sigma (x)=x}$.^[22]

Following the convention of omitting 1-cycles, one may interpret an individual cycle as a permutation which fixes all the elements not in the cycle (a cyclic permutation having only one cycle of length greater than 1). Then the list of disjoint cycles can be seen as the composition of these cyclic permutations. For example, the one-line permutation ${\displaystyle \sigma =265431}$ can be written in cycle notation as:

${\displaystyle \sigma =(126)(35)(4)=(126)(35).}$

This may be seen as the composition ${\displaystyle \sigma =\kappa _{1}\kappa _{2}}$ of cyclic permutations:

${\displaystyle \kappa _{1}=(126)=(126)(3)(4)(5),\quad \kappa _{2}=(35)=(35)(1)(2)(6).}$

While permutations in general do not commute, disjoint cycles do; for example:

${\displaystyle \sigma =(126)(35)=(35)(126).}$

Also, each cycle can be rewritten from a different starting point; for example,

${\displaystyle \sigma =(126)(35)=(261)(53).}$

Thus one may write the disjoint cycles of a given permutation in many different ways. A convenient feature of cycle notation is that inverting the permutation is given by reversing the order of the elements in each cycle. For example,

${\displaystyle \sigma ^{-1}=\left({\vphantom {A^{2}}}(126)(35)\right)^{-1}=(621)(53).}$

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, not omitting 1-cycles

For example,

(513)(6)(827)(94)

is a permutation of S = {1, 2, . . . , 9} in canonical cycle notation.^[23]

Richard Stanley calls this the "standard representation" of a permutation,^[24] and Martin Aigner uses "standard form".^[21] Sergey Kitaev also uses the "standard form" terminology, but reverses both choices; that is, each cycle lists its minimal element first, and the cycles are sorted in decreasing order of their minimal elements.^[25]

Composition of permutations

There are two ways to denote the composition of two permutations. In the most common notation, ${\displaystyle \sigma \cdot \tau }$ is the function that maps any element x to ${\displaystyle \sigma (\tau (x))}$. The rightmost permutation is applied to the argument first,^[26] because the argument is written to the right of the function.

A different rule for multiplying permutations comes from writing the argument to the left of the function, so that the leftmost permutation acts first.^[27][28][29] In this notation, the permutation is often written as an exponent, so σ acting on x is written x^σ; then the product is defined by ${\displaystyle x^{\sigma \cdot \tau }=(x^{\sigma })^{\tau }}$. This article uses the first definition, where the rightmost permutation is applied first.

The function composition operation satisfies the axioms of a group. It is associative, meaning ${\displaystyle (\rho \sigma )\tau =\rho (\sigma \tau )}$, and products of more than two permutations are usually written without parentheses. The composition operation also has an identity element (the identity permutation ${\displaystyle {\text{id}}}$), and each permutation ${\displaystyle \sigma }$ has an inverse ${\displaystyle \sigma ^{-1}}$ (its inverse function) with ${\displaystyle \sigma ^{-1}\sigma =\sigma \sigma ^{-1}={\text{id}}}$.

k-permutations of n

In older literature and elementary textbooks, a k-permutation of n (sometimes called a partial permutation, sequence without repetition, variation, or arrangement) means an ordered arrangement (list) of a k-element subset of an n-set.^{[lower-alpha 3][30][31]} The number of such k-permutations (k-arrangements) of ${\displaystyle n}$ is denoted variously by such symbols as ${\displaystyle P_{k}^{n}}$, ${\displaystyle _{n}P_{k}}$, ${\displaystyle ^{n}\!P_{k}}$, ${\displaystyle P_{n,k}}$, ${\displaystyle P(n,k)}$, or ${\displaystyle A_{n}^{k}}$,^[32] computed by the formula:^[33]

${\displaystyle P(n,k)=\underbrace {n\cdot (n-1)\cdot (n-2)\cdots (n-k+1)} _{k\ \mathrm {factors} }}$,

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

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

The product is well defined without the assumption that ${\displaystyle n}$ is a non-negative integer, and is of importance outside combinatorics as well; it is known as the Pochhammer symbol ${\displaystyle (n)_{k}}$ or as the ${\displaystyle k}$-th falling factorial power ${\displaystyle n^{\underline {k}}}$:

${\displaystyle P(n,k)={_{n}}P_{k}=(n)_{k}=n^{\underline {k}}.}$

This usage of the term permutation is closely associated with the term combination to mean a subset. A k-combination of a set S is a k-element subset of S: the elements of a combination are not ordered. Ordering the k-combinations of S in all possible ways produces 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:

${\displaystyle C(n,k)={\frac {P(n,k)}{P(k,k)}}={\frac {n^{\underline {k}}}{k!}}={\frac {n!}{(n-k)!\,k!}}.}$

These numbers are also known as binomial coefficients, usually denoted ${\displaystyle {\tbinom {n}{k}}}$:

${\displaystyle C(n,k)={_{n}}C_{k}={\binom {n}{k}}.}$

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 the usual sense. They are also called words over the alphabet S. If the set S has n elements, the number of k-tuples over S is ${\displaystyle n^{k}.}$ A formal language is a set of words obeying specified rules.

Permutations of multisets

Permutations without repetition on the left, with repetition to their right

If M is a finite multiset, then a multiset permutation is an ordered arrangement of elements of M in which each element appears a number of times equal exactly to its multiplicity in M. An anagram of a word having some repeated letters is an example of a multiset permutation.^{[lower-alpha 4]} If the multiplicities of the elements of M (taken in some order) are ${\displaystyle m_{1}}$, ${\displaystyle m_{2}}$, ..., ${\displaystyle m_{l}}$ 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,^[34]

${\displaystyle {n \choose m_{1},m_{2},\ldots ,m_{l}}={\frac {n!}{m_{1}!\,m_{2}!\,\cdots \,m_{l}!}}={\frac {\left(\sum _{i=1}^{l}{m_{i}}\right)!}{\prod _{i=1}^{l}{m_{i}!}}}.}$

For example, the number of distinct anagrams of the word MISSISSIPPI is:^[35]

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

A k-permutation of a multiset M is a sequence of k 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. If, however, the objects are arranged in a circular manner this distinguished ordering is weakened: there is no "first element" in the arrangement, as any element can be considered as the start. An arrangement of distinct objects in a circular manner is called a circular permutation.^{[36][lower-alpha 5]} These can be formally defined as equivalence classes of ordinary permutations of these objects, for the equivalence relation generated by moving the final element of the linear arrangement to its front.

Two circular permutations are equivalent if one can be rotated into the other. 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

There are (n – 1)! circular permutations of a set with n elements.

Cycle type

The cycles (including the fixed points) of a permutation ${\displaystyle \sigma }$ of a set with n elements partition that set; so the lengths of these cycles form an integer partition of n, which is called the cycle type (or sometimes cycle structure or cycle shape) of ${\displaystyle \sigma }$. There is a "1" in the cycle type for every fixed point of ${\displaystyle \sigma }$, a "2" for every transposition, and so on. The cycle type of ${\displaystyle \beta =(1\,2\,5\,)(\,3\,4\,)(6\,8\,)(\,7\,)}$ is ${\displaystyle (3,2,2,1).}$

This may also be written in a more compact form as [1¹2²3¹]. More precisely, the general form is ${\displaystyle [1^{\alpha _{1}}2^{\alpha _{2}}\dotsm n^{\alpha _{n}}]}$, where ${\displaystyle \alpha _{1},\ldots ,\alpha _{n}}$ are the numbers of cycles of respective length. The number of permutations of a given cycle type is^[38]

${\displaystyle {\frac {n!}{1^{\alpha _{1}}2^{\alpha _{2}}\dotsm n^{\alpha _{n}}\alpha _{1}!\alpha _{2}!\dotsm \alpha _{n}!}}}$.

The number of cycle types of a set with n elements equals the value of the partition function ${\displaystyle p(n)}$.

Polya's cycle index polynomial is a generating function which counts permutations by their cycle type.

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 ${\displaystyle \sigma }$ by another permutation ${\displaystyle \pi }$, which means forming the product ${\displaystyle \pi \sigma \pi ^{-1}}$. Here, ${\displaystyle \pi \sigma \pi ^{-1}}$ is the conjugate of ${\displaystyle \sigma }$ by ${\displaystyle \pi }$ and its cycle notation can be obtained by taking the cycle notation for ${\displaystyle \sigma }$ and applying ${\displaystyle \pi }$ to all the entries in it.^[39] It follows that two permutations are conjugate exactly when they have the same cycle type.

Order of a permutation

The order of a permutation ${\displaystyle \sigma }$ is the smallest positive integer m so that ${\displaystyle \sigma ^{m}=\mathrm {id} }$. It is the least common multiple of the lengths of its cycles. For example, the order of ${\displaystyle \sigma =(152)(34)}$ is ${\displaystyle {\text{lcm}}(3,2)=6}$.

Parity of a permutation

Every permutation of a finite set can be expressed as the product of transpositions.^[40] 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 ${\displaystyle \operatorname {sgn} \sigma }$, to each permutation. ${\displaystyle \operatorname {sgn} \sigma =+1}$ if ${\displaystyle \sigma }$ is even and ${\displaystyle \operatorname {sgn} \sigma =-1}$ if ${\displaystyle \sigma }$ is odd. Then for two permutations ${\displaystyle \sigma }$ and ${\displaystyle \pi }$

${\displaystyle \operatorname {sgn} (\sigma \pi )=\operatorname {sgn} \sigma \cdot \operatorname {sgn} \pi .}$

It follows that ${\displaystyle \operatorname {sgn} \left(\sigma \sigma ^{-1}\right)=+1.}$

The sign of a permutation is equal to the determinant of its permutation matrix (below).

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 ways to assign a permutation matrix to a permutation of {1, 2, ..., n}. One natural approach is to define ${\displaystyle L_{\sigma }}$ to be the linear transformation of ${\displaystyle \mathbb {R} ^{n}}$ which permutes the standard basis ${\displaystyle \{\mathbf {e} _{1},\ldots ,\mathbf {e} _{n}\}}$ by ${\displaystyle L_{\sigma }(\mathbf {e} _{j})=\mathbf {e} _{\sigma (j)}}$, and define ${\displaystyle M_{\sigma }}$ to be its matrix. That is, ${\displaystyle M_{\sigma }}$ has its j^th column equal to the n × 1 column vector ${\displaystyle \mathbf {e} _{\sigma (j)}}$: its (i, j) entry is to 1 if i = σ(j), and 0 otherwise. Since composition of linear mappings is described by matrix multiplication, it follows that this construction is compatible with composition of permutations:

${\displaystyle M_{\sigma }M_{\tau }=M_{\sigma \tau }}$.

For example, the one-line permutations ${\displaystyle \sigma =213,\ \tau =231}$ have product ${\displaystyle \sigma \tau =132}$, and the corresponding matrices are:

$${\displaystyle M_{\sigma }M_{\tau }={\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&0&1\\0&1&0\end{pmatrix}}=M_{\sigma \tau }.}$$

Composition of permutations corresponding to a multiplication of permutation matrices.

It is also common in the literature to find the inverse convention, where a permutation σ is associated to the matrix ${\displaystyle P_{\sigma }=(M_{\sigma })^{-1}=(M_{\sigma })^{T}}$ 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, ${\displaystyle P_{\sigma }P_{\tau }=P_{\tau \sigma }}$. In this correspondence, permutation matrices act on the right side of the standard ${\displaystyle 1\times n}$ row vectors ${\displaystyle ({\bf {e}}_{i})^{T}}$: ${\displaystyle ({\bf {e}}_{i})^{T}P_{\sigma }=({\bf {e}}_{\sigma (i)})^{T}}$.

The Cayley table on the right shows these matrices for permutations of 3 elements.

Ascents, descents, runs, exceedances, records

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, i is an ascent if ${\displaystyle \sigma (i)<\sigma (i{+}1)}$. For example, the permutation 3452167 has ascents (at positions) 1, 2, 5, and 6.

Similarly, a descent is a position i < n with ${\displaystyle \sigma (i)>\sigma (i{+}1)}$, so every i with ${\displaystyle 1\leq i<n}$ is either an ascent or a descent.

An ascending run of a permutation is a nonempty increasing contiguous subsequence 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 obtained by omitting some of the values of the one-line notation. 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.^[41]

The number of permutations of n with k ascents is (by definition) the Eulerian number ${\displaystyle \textstyle \left\langle {n \atop k}\right\rangle }$; this is also the number of permutations of n with k descents. Some authors however define the Eulerian number ${\displaystyle \textstyle \left\langle {n \atop k}\right\rangle }$ as the number of permutations with k ascending runs, which corresponds to k − 1 descents.^[42]

An exceedance of a permutation σ₁σ₂...σ_n is an index j such that σ_j > j. If the inequality is not strict (that is, σ_j ≥ j), then j is called a weak exceedance. The number of n-permutations with k exceedances coincides with the number of n-permutations with k descents.^[43]

A record or left-to-right maximum of a permutation σ is an element i such that σ(j) < σ(i) for all j < i.

Foata's transition lemma

Foata's fundamental bijection transforms a permutation ${\displaystyle \sigma }$ with a given canonical cycle form into the permutation ${\displaystyle f(\sigma )={\hat {\sigma }}}$ whose one-line notation has the same sequence of elements with parentheses removed.^[24][44] For example:

${\displaystyle \sigma =(513)(6)(827)(94)={\begin{pmatrix}1&2&3&4&5&6&7&8&9\\3&7&5&9&1&6&8&2&4\end{pmatrix}},}$ ${\displaystyle {\hat {\sigma }}=513682794={\begin{pmatrix}1&2&3&4&5&6&7&8&9\\5&1&3&6&8&2&7&9&4\end{pmatrix}}.}$

Here the first element in each canonical cycle of ${\displaystyle \sigma }$ becomes a record (left-to-right maximum) of ${\displaystyle {\hat {\sigma }}}$. Given ${\displaystyle {\hat {\sigma }}}$, one may find its records and insert parentheses to construct the inverse transformation ${\displaystyle \sigma =f^{-1}({\hat {\sigma }})}$. Underlining the records in the above example: ${\displaystyle {\hat {\sigma }}={\underline {5}}\,1\,3\,{\underline {6}}\,{\underline {8}}\,2\,7\,{\underline {9}}\,4}$, which allows the reconstruction of the cycles of ${\displaystyle \sigma }$.

The following table shows ${\displaystyle {\hat {\sigma }}}$ and ${\displaystyle \sigma }$ for the six permutations of S = {1,2,3}, with the bold text on each side showing the notation used in the bijection: one-line notation for ${\displaystyle {\hat {\sigma }}}$ and canonical cycle notation for ${\displaystyle \sigma }$. Note that the first 3 permutations are mapped to themselves under the bijection.

${\displaystyle {\hat {\sigma }}=f(\sigma )}$	${\displaystyle \sigma =f^{-1}({\hat {\sigma }})}$
${\displaystyle \mathbf {123} =(\,1\,)(\,2\,)(\,3\,)}$	${\displaystyle 123=\mathbf {(\,1\,)(\,2\,)(\,3\,)} }$
${\displaystyle \mathbf {132} =(\,1\,)(\,3\,2\,)}$	${\displaystyle 132=\mathbf {(\,1\,)(\,3\,2\,)} }$
${\displaystyle \mathbf {213} =(\,2\,1\,)(\,3\,)}$	${\displaystyle 213=\mathbf {(\,2\,1\,)(\,3\,)} }$
${\displaystyle \mathbf {231} =(\,3\,1\,2\,)}$	${\displaystyle 321=\mathbf {(\,2\,)(\,3\,1\,)} }$
${\displaystyle \mathbf {312} =(\,3\,2\,1\,)}$	${\displaystyle 231=\mathbf {(\,3\,1\,2\,)} }$
${\displaystyle \mathbf {321} =(\,2\,)(\,3\,1\,)}$	${\displaystyle 312=\mathbf {(\,3\,2\,1\,)} }$

As a first corollary, the number of n-permutations with exactly k records is equal to the number of n-permutations with exactly k cycles: this last number is the signless Stirling number of the first kind, ${\displaystyle c(n,k)}$. Furthermore, Foata's mapping takes an n-permutation with k weak exceedances to an n-permutation with k − 1 ascents.^[44] For example, (2)(31) = 321 has k = 2 weak exceedances (at index 1 and 2), whereas f(321) = 231 has k − 1 = 1 ascent (at index 1; that is, from 2 to 3).

Inversions

In the 15 puzzle the goal is to get the squares in ascending order. Initial positions which have an odd number of inversions are impossible to solve.^[45]

An inversion of a permutation σ is a pair (i, j) of positions where the entries of a permutation are in the opposite order: ${\displaystyle i<j}$ and ${\displaystyle \sigma (i)>\sigma (j)}$.^[46] Thus a descent is an inversion at two adjacent positions. For example, σ = 23154 has (i, j) = (1,3), (2,3), and (4,5), where (σ(i), σ(j)) = (2,1), (3,1), and (5,4).

Sometimes an inversion is defined as the pair of values (σ(i), σ(j)); this makes no difference for the number of inversions, and the reverse pair (σ(j), σ(i)) is 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 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; 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.^[47] This is the coefficient of ${\displaystyle q^{k}}$ in the expansion of the product

$${\displaystyle [n]_{q}!=\prod _{m=1}^{n}\sum _{i=0}^{m-1}q^{i}=1\left(1+q\right)\left(1+q+q^{2}\right)\cdots \left(1+q+q^{2}+\cdots +q^{n-1}\right),}$$

The notation ${\displaystyle [n]_{q}!}$ denotes the q-factorial. This expansion commonly appears in the study of necklaces.

Let ${\displaystyle \sigma \in S_{n},i,j\in \{1,2,\dots ,n\}}$ such that ${\displaystyle i<j}$ and ${\displaystyle \sigma (i)>\sigma (j)}$. In this case, say the weight of the inversion ${\displaystyle (i,j)}$ is ${\displaystyle \sigma (i)-\sigma (j)}$. Kobayashi (2011) proved the enumeration formula

$${\displaystyle \sum _{i<j,\sigma (i)>\sigma (j)}(\sigma (i)-\sigma (j))=|\{\tau \in S_{n}\mid \tau \leq \sigma ,\tau {\text{ is bigrassmannian}}\}}$$

where ${\displaystyle \leq }$ denotes Bruhat order in the symmetric groups. This graded partial order often appears in the context of Coxeter groups.

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, ..., d₂, d₁, where d_i 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 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 (n − 1)!, (n − 2)!, ..., 2!, 1!. The second step interprets this sequence as a Lehmer code or (almost equivalently) as an inversion table.

Rothe diagram for ${\displaystyle \sigma =(6,3,8,1,4,9,7,2,5)}$

σi i	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 d_n represents the choice made for the first term σ₁, the number d_n−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 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.^[48]

Both encodings can be visualized by an n by n Rothe diagram^[49] (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, ..., 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 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, ..., d₂, d₁ into the corresponding permutation, one can traverse the numbers from d₁ to d_n while inserting the elements of S from largest to smallest into an initially empty sequence; at the step using the number d from the inversion table, the element from S inserted into the sequence at the point where it is preceded by d elements already present. Alternatively one could process the numbers from the inversion table and the elements of S both in the opposite order, starting with a row of n empty slots, and at each step place the element from S into the empty slot that is preceded by d other empty slots.

Converting successive natural numbers to the factorial number system produces those sequences in lexicographic order (as is the case with any mixed radix number system), and further converting them to permutations preserves the lexicographic ordering, provided the Lehmer code interpretation is used (using inversion tables, one gets a different ordering, where one starts by comparing permutations by the place of their entries 1 rather than by the value of their first entries). The sum of the numbers in the factorial number system representation gives the number of inversions of the permutation, and the parity of that sum gives the signature of the permutation. Moreover, the positions of the zeroes in the inversion table give the values of left-to-right maxima of the permutation (in the example 6, 8, 9) while the positions of the zeroes in the Lehmer code are the positions of the right-to-left minima (in the example positions the 4, 8, 9 of the values 1, 2, 5); this allows computing the distribution of such extrema among all permutations. A permutation with Lehmer code d_n, d_n−1, ..., d₂, d₁ has an ascent n − i if and only if d_i ≥ d_i+1.

Algorithms to generate permutations

In computing it may be required to generate permutations of a given sequence of values. The methods best adapted to do this depend on whether one wants some randomly chosen permutations, or all permutations, and in the latter case if a specific ordering is required. Another question is whether possible equality among entries in the given sequence is to be taken into account; if so, one should only generate distinct multiset permutations of the sequence.

An obvious way to generate permutations of n is to generate values for the Lehmer code (possibly using the factorial number system representation of integers up to n!), and convert those into the corresponding permutations. However, the latter step, while straightforward, is hard to implement efficiently, because it requires n operations each of selection from a sequence and deletion from it, at an arbitrary position; of the obvious representations of the sequence as an array or a linked list, both require (for different reasons) about 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.

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

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

Generation with minimal changes

Main articles: Steinhaus–Johnson–Trotter algorithm and Heap's algorithm

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.^[53]

An alternative to Steinhaus–Johnson–Trotter is Heap's algorithm,^[55] said by Robert Sedgewick in 1977 to be the fastest algorithm of generating permutations in applications.^[52]

The following figure shows the output of all three aforementioned algorithms for generating all permutations of length ${\displaystyle n=4}$, and of six additional algorithms described in the literature.

Ordering of all permutations of length ${\displaystyle n=4}$ generated by different algorithms. The permutations are color-coded, where   1,   2,   3,   4.^[56]

1. Lexicographic ordering;
2. Steinhaus–Johnson–Trotter algorithm;
3. Heap's algorithm;
4. Ehrlich's star-transposition algorithm:^[53] in each step, the first entry of the permutation is exchanged with a later entry;
5. Zaks' prefix reversal algorithm:^[57] in each step, a prefix of the current permutation is reversed to obtain the next permutation;
6. Sawada-Williams' algorithm:^[58] each permutation differs from the previous one either by a cyclic left-shift by one position, or an exchange of the first two entries;
7. Corbett's algorithm:^[59] each permutation differs from the previous one by a cyclic left-shift of some prefix by one position;
8. Single-track ordering:^[60] each column is a cyclic shift of the other columns;
9. Single-track Gray code:^[60] each column is a cyclic shift of the other columns, plus any two consecutive permutations differ only in one or two transpositions.

Applications

Permutations are used in the interleaver component of the error detection and correction algorithms, such as turbo codes, for example 3GPP Long Term Evolution mobile telecommunication standard uses these ideas (see 3GPP technical specification 36.212^[62]). 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.^[63]