Permutation

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Each of the six rows is a different permutation of three distinct balls

In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.

Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1, 2, 3}, namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory.

Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences.

The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n.

Technically, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f(s). For example, the permutation (3, 1, 2) mentioned above is described by the function ${\displaystyle \alpha }$ defined as

${\displaystyle \alpha (1)=3,\alpha (2)=1,\alpha (3)=2}$.

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 ${\displaystyle \{1,2,\dots ,n\}}$ that are considered for studying permutations.

In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.

In the popular puzzle Rubik's cube invented in 1974 by Ernő Rubik, each turn of the puzzle faces creates a permutation of the surface colors.

History

Permutations called hexagrams were used in China in the I Ching (Pinyin: Yi Jing) as early as 1000 BC.

In Greece, Plutarch wrote that Xenocrates of Chalcedon (396–314 BC) discovered the number of different syllables possible in the Greek language. This would have been the first attempt on record to solve a difficult problem in permutations and combinations.

Al-Khalil (717–786), an Arab mathematician and cryptographer, wrote the Book of Cryptographic Messages. It contains the first use of permutations and combinations, to list all possible Arabic words with and without vowels.

The rule to determine the number of permutations of n objects was known in Indian culture around 1150 AD. The Lilavati by the Indian mathematician Bhaskara II contains a passage that translates as follows:

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 played an important role in the cryptanalysis of the Enigma machine, a cipher device used by Nazi Germany during World War II. In particular, one important property of permutations, namely, that two permutations are conjugate exactly when they have the same cycle type, was used by cryptologist Marian Rejewski to break the German Enigma cipher in turn of years 1932-1933.

Permutations without repetitions

The simplest example of permutations is permutations without repetitions where we consider the number of possible ways of arranging n items into n places. The factorial has special application in defining the number of permutations in a set which does not include repetitions. The number n!, read "n factorial", is precisely the number of ways we can rearrange n things into a new order. For example, if we have three fruits: an orange, apple and pear, we can eat them in the order mentioned, or we can change them (for example, an apple, a pear then an orange). The exact number of permutations is then ${\displaystyle 3!=1\cdot 2\cdot 3=6}$. 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 ${\displaystyle nPk}$ (which reads "n permute k"), and is equal to the number ${\displaystyle n(n-1)\cdots (n-k+1)}$ (also written as ${\displaystyle n!/(n-k)!}$).

Definition

In mathematics texts it is customary to denote permutations using lowercase Greek letters. Commonly, either ${\displaystyle \alpha }$ and ${\displaystyle \beta ,}$ or ${\displaystyle \sigma ,\tau }$ and ${\displaystyle \pi }$ are used.

Permutations can be defined as bijections from a set S onto itself. All permutations of a set with n elements form a symmetric group, denoted ${\displaystyle S_n}$, where the group operation is function composition. Thus for two permutations, ${\displaystyle \pi }$ and ${\displaystyle \sigma }$ in the group ${\displaystyle S_n}$, the four group axioms hold:

1. Closure: If ${\displaystyle \pi }$ and ${\displaystyle \sigma }$ are in ${\displaystyle S_n}$ then so is ${\displaystyle \pi \sigma .}$
2. Associativity: For any three permutations ${\displaystyle \pi ,\sigma ,\tau \in S_n}$, ${\displaystyle (\pi \sigma )\tau =\pi (\sigma \tau ).}$
3. Identity: There is an identity permutation, denoted ${\displaystyle \mathrm{id}}$ and defined by ${\displaystyle \mathrm{id}(x)=x}$ for all ${\displaystyle x\in S}$. For any ${\displaystyle \sigma \in S_n}$, ${\displaystyle \mathrm{id}\sigma =\sigma \mathrm{id}=\sigma .}$
4. Invertibility: For every permutation ${\displaystyle \pi \in S_n}$, there exists an inverse permutation ${\displaystyle {\pi }^{-1}\in S_n}$, so that ${\displaystyle \pi {\pi }^{-1}={\pi }^{-1}\pi =\mathrm{id}.}$

In general, composition of two permutations is not commutative, that is, ${\displaystyle \pi \sigma \ne \sigma \pi .}$

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, which are found by repeatedly tracing the application of the permutation on some elements. For example, the permutation ${\displaystyle \sigma }$ defined by ${\displaystyle \sigma (7)=7}$ has a 1-cycle, ${\displaystyle (7)}$ while the permutation ${\displaystyle \pi }$ defined by ${\displaystyle \pi (2)=3}$ and ${\displaystyle \pi (3)=2}$ has a 2-cycle ${\displaystyle (23)}$ (for details on the syntax, see § Cycle notation below). In general, a cycle of length k, that is, consisting of k elements, is called a k-cycle.

An element in a 1-cycle ${\displaystyle (x)}$ is called a fixed point of the permutation. A permutation with no fixed points is called a derangement. 2-cycles are called transpositions; such permutations merely exchange two elements, leaving the others fixed.

Notations

Since writing permutations elementwise, that is, as piecewise functions, is cumbersome, several notations have been invented to represent them more compactly. Cycle notation is a popular choice for many mathematicians due to its compactness and the fact that it makes a permutation's structure transparent. It is the notation used in this article unless otherwise specified, but other notations are still widely used, especially in application areas.

Two-line notation

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

${\displaystyle \sigma =\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 2& 5& 4& 3& 1\end{array}\right);}$

this means that σ satisfies σ(1) = 2, σ(2) = 5, σ(3) = 4, σ(4) = 3, and σ(5) = 1. The elements of S may appear in any order in the first row. This permutation could also be written as:

${\displaystyle \sigma =\left(\begin{array}{ccccc}3& 2& 5& 1& 4\\ 4& 5& 1& 2& 3\end{array}\right),}$

or

${\displaystyle \sigma =\left(\begin{array}{ccccc}5& 1& 4& 3& 2\\ 1& 2& 3& 4& 5\end{array}\right).}$

One-line notation

If there is a "natural" order for the elements of S, say ${\displaystyle x_1,x_2,\dots ,x_n}$, then one uses this for the first row of the two-line notation:

${\displaystyle \sigma =\left(\begin{array}{ccccc}{x}_1& x_2& x_3& \cdots & x_n\\ \sigma (x_1)& \sigma (x_2)& \sigma (x_3)& \cdots & \sigma (x_n)\end{array}\right).}$

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

${\displaystyle (\sigma (x_1)\sigma (x_2)\sigma (x_3)\cdots \sigma (x_n))}$,

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 representation of a permutation. The example above would then be 2 5 4 3 1 since the natural order 1 2 3 4 5 would be assumed for the first row. (It is typical to use commas to separate these entries only if some have two or more digits.) This form is more compact, and is common in elementary combinatorics and computer science. It is especially useful in applications where the elements of S or the permutations are to be compared as larger or smaller.

Cycle notation

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 ${\displaystyle \sigma }$ in cycle notation, one proceeds as follows:

1. Write an opening bracket then select an arbitrary element x of ${\displaystyle S}$ and write it down: ${\displaystyle (x}$
2. Then trace the orbit of x; that is, write down its values under successive applications of ${\displaystyle \sigma }$: ${\displaystyle (x\sigma (x)\sigma (\sigma (x))\dots }$
3. Repeat until the value returns to x and write down a closing parenthesis rather than x: ${\displaystyle (x\sigma (x)\sigma (\sigma (x))\dots )}$
4. Now continue with an element y of S, not yet written down, and proceed in the same way: ${\displaystyle (x\sigma (x)\sigma (\sigma (x))\dots )(y\dots )}$
5. Repeat until all elements of S are written in cycles.

So the permutation 2 5 4 3 1 (in one-line notation) could be written as (125)(34) in cycle notation.

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

$${\displaystyle (125)(34)=(34)(125).}$$

In addition, each cycle can be written in different ways, by choosing different starting points; for example,

$${\displaystyle (125)(34)=(512)(34)=(251)(43).}$$

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 ${\displaystyle \sigma (x)=x}$. 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,

$${\displaystyle [(125)(34){]}^{-1}=(521)(43).}$$

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. ${\displaystyle \sigma \cdot \pi }$ is the function that maps any element x of the set to ${\displaystyle \sigma (\pi (x))}$. The rightmost permutation is applied to the argument first, because of the way the function application is written.

Since function composition is associative, so is the composition operation on permutations: ${\displaystyle (\sigma \pi )\tau =\sigma (\pi \tau )}$. 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 x^σ·π = (x^σ)^π. However this gives a different rule for multiplying permutations; this article uses the definition where the rightmost permutation is applied first.

Other uses of the term permutation

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 ${\displaystyle k}$-permutations 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 ${\displaystyle A_n^k}$ (the ${\displaystyle A}$ stands for "arrangements"), and its value is given by the product

${\displaystyle P(n,k)=\underset{k\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}{\underbrace{n\cdot (n-1)\cdot (n-2)\cdots (n-k+1)}}}$,

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 ${\displaystyle n^{\underset{\_}{k}}}$ of ${\displaystyle n}$.

This usage of the term permutation is closely related to the term combination. A k-element combination of an n-set S is a k element subset of S, the elements of which are not ordered. By taking all the k element subsets of S and ordering each of them in all possible ways, we obtain all the k-permutations of S. The number of k-combinations of an n-set, C(n,k), is therefore related to the number of k-permutations of n by:

${\displaystyle C(n,k)=\frac{P(n,k)}{P(k,k)}=\frac{\frac{n!}{(n-k)!}}{\frac{k!}{0!}}=\frac{n!}{(n-k)!k!}.}$

These numbers are also known as binomial coefficients and are denoted by ${\displaystyle (\genfrac{}{}{0ex}{}{n}{k})}$.

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 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 ${\displaystyle n^k.}$ 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

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. 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,

${\displaystyle (\genfrac{}{}{0ex}{}{n}{m_1,m_2,\dots ,m_l})=\frac{n!}{m_1!m_2!\cdots m_l!}=\frac{\left(\sum _{i=1}^l{m}_i\right)!}{\prod _{i=1}^l{m}_i!}.}$

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

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

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 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 n distinct objects is n!.

The number of n-permutations with k disjoint cycles is the signless Stirling number of the first kind, denoted by c(n, k).

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 ${\displaystyle \beta =(125)(34)(68)(7)}$ is ${\displaystyle (3,2,2,1).}$

This may also be written in a more compact form as [1¹2²3¹]. More precisely, the general form is ${\displaystyle [1^{{\alpha }_1}{2}^{{\alpha }_2}\cdots n^{{\alpha }_n}]}$, where ${\displaystyle {\alpha }_1,\dots ,{\alpha }_n}$ are the numbers of cycles of respective length. The number of permutations of a given cycle type is

${\displaystyle \frac{n!}{1^{{\alpha }_1}{2}^{{\alpha }_2}\cdots n^{{\alpha }_n}{\alpha }_1!{\alpha }_2!\cdots {\alpha }_n!}}$.

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

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. It follows that two permutations are conjugate exactly when they have the same cycle type.

Permutation order

The order of a permutation ${\displaystyle \sigma }$ is the smallest positive integer m so that ${\displaystyle {\sigma }^m=\mathrm{i}\mathrm{d}}$. It is the least common multiple of its cycles lengths. For example, the order of ${\displaystyle (132)(45)}$ is ${\displaystyle 2\cdot 3=6}$.

Parity of a permutation

Main article: 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 ${\displaystyle \mathrm{sgn}\sigma }$, to each permutation. ${\displaystyle \mathrm{sgn}\sigma =+1}$ if ${\displaystyle \sigma }$ is even and ${\displaystyle \mathrm{sgn}\sigma =-1}$ if ${\displaystyle \sigma }$ is odd. Then for two permutations ${\displaystyle \sigma }$ and ${\displaystyle \pi }$

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

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

Matrix representation

Main article: Permutation matrix

A permutation matrix is an n × n matrix that has exactly one entry 1 in each column and in each row, and all other entries are 0. There are several different conventions that one can use to assign a permutation matrix to a permutation of {1, 2, ..., n}. One natural approach is to associate to the permutation σ the matrix ${\displaystyle M_{\sigma }}$ 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, ${\displaystyle M_{\sigma }{M}_{\pi }=M_{\sigma \circ \pi }}$ for all permutations σ and π. Second, if ${\displaystyle {\mathbf{e}}_i}$ represents the standard ${\displaystyle n\times 1}$ column vector (the vector with ith entry equal to 1 and all other entries equal to 0), then ${\displaystyle M_{\sigma }{\mathbf{e}}_i={\mathbf{e}}_{\sigma (i)}}$.

For example, with this convention, the matrix associated to the permutation ${\displaystyle \sigma (1,2,3)=(2,1,3)}$ is ${\displaystyle \left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)}$ and the matrix associated to the permutation ${\displaystyle \pi (1,2,3)=(2,3,1)}$ is ${\displaystyle \left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)}$. Then the composition of permutations is ${\displaystyle (\sigma \circ \pi )(1,2,3)=\sigma (2,3,1)=(1,3,2)}$, and the corresponding matrix product is

$${\displaystyle M_{(2,1,3)}{M}_{(2,3,1)}=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)=M_{(1,3,2)}.}$$

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, ${\displaystyle P_{\sigma }{P}_{\pi }=P_{\pi \circ \sigma }}$ for all permutations σ and π. In this correspondence, permutation matrices act by permuting indices of standard ${\displaystyle 1\times n}$ row vectors ${\displaystyle ({\mathbf{e}}_i{)}^T}$: one has ${\displaystyle ({\mathbf{e}}_i{)}^T{P}_{\sigma }=({\mathbf{e}}_{\sigma (i)}{)}^T}$.

The Cayley table 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 so that any two elements can be compared. The set {1, 2, ..., n} is totally ordered by the usual "≤" relation and so it is the most frequently used set in these applications, but in general, any totally ordered set will do. In these applications, the ordered arrangement view of a permutation is needed to talk about the positions in a permutation.

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

Ascents, descents, runs and exceedances

An ascent of a permutation σ of n is any position i < n where the following value is bigger than the current one. That is, if σ = σ₁σ₂...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 ${\displaystyle 1\le i<n}$ either is an ascent or is a descent of σ.

An ascending run of a permutation is a nonempty increasing contiguous subsequence of the permutation that cannot be extended at either end; it corresponds to a maximal sequence of successive ascents (the latter may be empty: between two successive descents there is still an ascending run of length 1). By contrast an increasing subsequence of a permutation is not necessarily contiguous: it is an increasing sequence of elements obtained from the permutation by omitting the values at some positions. For example, the permutation 2453167 has the ascending runs 245, 3, and 167, while it has an increasing subsequence 2367.

If a permutation has k − 1 descents, then it must be the union of k ascending runs.

The number of permutations of n with k ascents is (by definition) the Eulerian number ${\displaystyle ⟨\genfrac{}{}{0ex}{}{n}{k}⟩}$; this is also the number of permutations of n with k descents. Some authors however define the Eulerian number ${\displaystyle ⟨\genfrac{}{}{0ex}{}{n}{k}⟩}$ as the number of permutations with k ascending runs, which corresponds to k − 1 descents.

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.

Foata's transition lemma

There is a relationship between the one-line notation and the canonical cycle notation. Consider the permutation ${\displaystyle (2)(31)}$ in canonical cycle notation; if we simply remove the parentheses, we obtain the permutation ${\displaystyle 231}$ 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 ${\displaystyle f(p)=q}$ be the parentheses-erasing transformation which returns ${\displaystyle q}$ in one-line notation when given ${\displaystyle p}$ in canonical cycle notation. As stated, ${\displaystyle f}$ operates by simply removing all parentheses. The operation of the inverse transformation, ${\displaystyle f^{-1}(q)=p}$, which returns ${\displaystyle p}$ in canonical cycle notation when given ${\displaystyle q}$ in one-line notation, is a bit less intuitive. Given the one-line notation ${\displaystyle q=q_1{q}_2\cdots q_n}$, the first cycle of ${\displaystyle p}$ in canonical cycle notation must start with ${\displaystyle q_1}$. As long as the subsequent elements are smaller than ${\displaystyle q_1}$, we are in the same cycle of ${\displaystyle p}$. The second cycle of ${\displaystyle p}$ starts at the smallest index ${\displaystyle j}$ such that ${\displaystyle q_j>q_1}$. In other words, ${\displaystyle q_j}$ 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 ${\displaystyle q=312548976}$, 5 is the first element larger than the starting element 3, so the first cycle of ${\displaystyle p}$ must be ${\displaystyle (312)}$. Then 8 is the next element larger than 5, so the second cycle is ${\displaystyle (54)}$. Since 9 is larger than 8, ${\displaystyle (8)}$ is a cycle by itself. Finally, 9 is larger than all the remaining elements to its right, so the last cycle is ${\displaystyle (976)}$. Concatenating these 4 cycles gives ${\displaystyle p=(312)(54)(8)(976)}$ in canonical cycle notation.

The following table shows both ${\displaystyle q}$ and ${\displaystyle p}$ for the six permutations of ${\displaystyle 123}$. The bold side of each equality shows the permutation using its designated notation (one-line notation for ${\displaystyle q}$ and canonical cycle notation for ${\displaystyle p}$) 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).

${\displaystyle q=f(p)}$	${\displaystyle p=f^{-1}(q)}$
${\displaystyle \mathbf{123}=(1)(2)(3)}$	${\displaystyle 123=\mathbf{(}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{3}\mathbf{)}}$
${\displaystyle \mathbf{132}=(1)(32)}$	${\displaystyle 132=\mathbf{(}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{3}\mathbf{2}\mathbf{)}}$
${\displaystyle \mathbf{213}=(21)(3)}$	${\displaystyle 213=\mathbf{(}\mathbf{2}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{3}\mathbf{)}}$
${\displaystyle \mathbf{231}=(312)}$	${\displaystyle 321=\mathbf{(}\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{3}\mathbf{1}\mathbf{)}}$
${\displaystyle \mathbf{312}=(321)}$	${\displaystyle 231=\mathbf{(}\mathbf{3}\mathbf{1}\mathbf{2}\mathbf{)}}$
${\displaystyle \mathbf{321}=(2)(31)}$	${\displaystyle 312=\mathbf{(}\mathbf{3}\mathbf{2}\mathbf{1}\mathbf{)}}$

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, ${\displaystyle c(n,k)}$. Furthermore, Foata's mapping takes an n-permutation with k-weak exceedances to an n-permutations with k − 1 ascents. For example, (2)(31) = 321 has two weak exceedances (at index 1 and 2), whereas f(321) = 231 has one ascent (at index 1; that is, from 2 to 3).

Inversions

Main article: Inversion (discrete mathematics)

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.

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 ${\displaystyle {\sigma }_i>{\sigma }_j}$. So a descent is just an inversion at two adjacent positions. For example, the permutation σ = 23154 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 (σ_i,σ_j) 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 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, it is the coefficient of X^k in the expansion of the product

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

which is also known (with q substituted for X) as the q-factorial [n]_q! . The expansion of the product appears in Necklace (combinatorics).

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

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

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

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, ..., 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. 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, ..., 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.

Random generation of permutations

Main article: Fisher–Yates shuffle

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 d₁,d₂,...,d_n satisfying 0 ≤ d_i < i (since d₁ is always zero it may be omitted) and to convert it to a permutation through a bijective correspondence. For the latter correspondence one could interpret the (reverse) sequence as a Lehmer code, and this gives a generation method first published in 1938 by Ronald Fisher and Frank Yates. While at the time computer implementation was not an issue, this method suffers from the difficulty sketched above to convert from Lehmer code to permutation efficiently. This can be remedied by using a different bijective correspondence: after using d_i to select an element among i remaining elements of the sequence (for decreasing values of i), rather than removing the element and compacting the sequence by shifting down further elements one place, one swaps the element with the final remaining element. Thus the elements remaining for selection form a consecutive range at each point in time, even though they may not occur in the same order as they did in the original sequence. The mapping from sequence of integers to permutations is somewhat complicated, but it can be seen to produce each permutation in exactly one way, by an immediate induction. When the selected element happens to be the final remaining element, the swap operation can be omitted. This does not occur sufficiently often to warrant testing for the condition, but the final element must be included among the candidates of the selection, to guarantee that all permutations can be generated.

The resulting algorithm for generating a random permutation of a[0], a[1], ..., a[n − 1] can be described as follows in pseudocode:

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

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

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

If 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.

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

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

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

Following this algorithm, the next lexicographic permutation will be [1, 3, 2, 4], and the 24th permutation will be [4, 3, 2, 1] at which point a[k] < a[k + 1] does not exist, indicating that this is the last permutation.

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

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.

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 ${\displaystyle n=4}$, 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.

1. Lexicographic ordering;
2. Steinhaus–Johnson–Trotter algorithm;
3. Heap's algorithm;
4. Ehrlich's star-transposition algorithm: in each step, the first entry of the permutation is exchanged with a later entry;
5. Zaks' prefix reversal algorithm: in each step, a prefix of the current permutation is reversed to obtain the next permutation;
6. 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;
7. Corbett's algorithm: each permutation differs from the previous one by a cyclic left-shift of some prefix by one position;
8. Single-track ordering: each column is a cyclic shift of the other columns;
9. 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 {1, 2, ..., 2n} is a cyclic permutation (with no fixed points) such that the digits in the cyclic notation form alternate between odd and even integers. Meandric permutations are useful in the analysis of RNA secondary structure. Not all alternate permutations are meandric. A modification of Heap's algorithm has been used to generate all alternate permutations of order n (that is, of length 2n) without generating all (2n)! permutations. 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).

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

Applications

Permutations are used in the interleaver component of the error detection and correction algorithms, such as turbo codes, for example 3GPP Long Term Evolution mobile telecommunication standard uses these ideas (see 3GPP technical specification 36.212). Such applications raise the question of fast generation of permutations satisfying certain desirable properties. One of the methods is based on the permutation polynomials. Also as a base for optimal hashing in Unique Permutation Hashing.

Electron microscope

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Electron_microscope

A modern transmission electron microscope

An image of an ant in a scanning electron microscope

An electron microscope is a microscope that uses a beam of electrons as a source of illumination. They use electron optics that are analogous to the glass lenses of an optical light microscope. As the wavelength of an electron can be up to 100,000 times shorter than that of visible light, electron microscopes have a higher resolution of about 0.1 nm, which compares to about 200 nm for light microscopes. Electron microscope may refer to:

• Transmission electron microscopy (TEM) where swift electrons go through a thin sample
• Scanning transmission electron microscopy (STEM) is similar to TEM with a scanned electron probe
• Scanning electron microscope (SEM) is similar the STEM, but with thick samples
• Electron microprobe similar to a SEM, but more for chemical analysis
• Ultrafast scanning electron microscopy, version of SEM that can operate very fast
• Low-energy electron microscopy (LEEM), used to image surfaces
• Photoemission electron microscopy (PEEM) is similar to LEEM using electrons produced at surfaces by photons

Additional details can be found in the above. This articles contains some general information mainly about transmission electron microscopes.

History

See also: Transmission electron microscopy § History

Reproduction of an early electron microscope constructed by Ernst Ruska

Many developments laid the groundwork of the electron optics used in microscopes. One significant step was the work of Hertz in 1883 who made a cathode-ray tube with electrostatic and magnetic deflection, demonstrating manipulation of the direction of an electron beam. Others were focusing of the electrons by an axial magnetic field by Emil Wiechert in 1899, improved oxide-coated cathodes which produced more electrons by Arthur Wehnelt in 1905 and the development of the electromagnetic lens in 1926 by Hans Busch. According to Dennis Gabor, the physicist Leó Szilárd tried in 1928 to convince him to build an electron microscope, for which Szilárd had filed a patent.

To this day the issue of who invented the transmission electron microscope is controversial. In 1928, at the Technical University of Berlin, Adolf Matthias (Professor of High Voltage Technology and Electrical Installations) appointed Max Knoll to lead a team of researchers to advance research on electron beams and cathode-ray oscilloscopes. The team consisted of several PhD students including Ernst Ruska. In 1931, Max Knoll and Ernst Ruska successfully generated magnified images of mesh grids placed over an anode aperture. The device, a replicate of which is shown in the figure, used two magnetic lenses to achieve higher magnifications, the first electron microscope. (Max Knoll died in 1969, so did not receive a share of the Nobel Prize in 1986.)

Apparently independent of this effort was work at Siemens-Schuckert by Reinhold Rüdenberg. According to patent law (U.S. Patent No. 2058914 and 2070318, both filed in 1932), he is the inventor of the electron microscope, but it is not clear when he had a working instrument. He stated in a very brief article in 1932 that Siemens had been working on this for some years before the patents were filed in 1932, claiming that his effort was parallel to the university development. He died in 1961, so similar to Max Knoll, was not eligible for a share of the Nobel Prize.

In the following year, 1933, Ruska and Knoll built the first electron microscope that exceeded the resolution attainable with an optical (light) microscope. Four years later, in 1937, Siemens financed the work of Ernst Ruska and Bodo von Borries, and employed Helmut Ruska, Ernst's brother, to develop applications for the microscope, especially with biological specimens. Also in 1937, Manfred von Ardenne pioneered the scanning electron microscope. Siemens produced the first commercial electron microscope in 1938. The first North American electron microscopes were constructed in the 1930s, at the Washington State University by Anderson and Fitzsimmons  and at the University of Toronto by Eli Franklin Burton and students Cecil Hall, James Hillier, and Albert Prebus. Siemens produced a transmission electron microscope (TEM) in 1939. Although current transmission electron microscopes are capable of two million-power magnification, as scientific instruments they remain similar but with improved optics.

Wavelength

Diagram illustrating the phenomena resulting from the interaction of energetic electrons with matter

See also: Electron diffraction

In a typical electron gun, individual electrons, which have an elementary charge ${\displaystyle e}$ (about ${\displaystyle -1.6\times {10}^{-19}}$ coulombs) and a mass ${\displaystyle m}$ (about ${\displaystyle 9.1\times {10}^{-31}}$ kg), with a potential of ${\displaystyle V}$ volts, have an energy amount of ${\displaystyle e\cdot V}$ joules. The wavelength is

${\displaystyle \lambda =\frac{hc}{\sqrt{eV(2m{c}^2+eV)}}}$,

where ${\displaystyle c}$ is the speed of light in vacuum (about ${\displaystyle c=3\times {10}^8}$ m/s). See electron diffraction for a full explanation.

Types

Transmission electron microscope (TEM)

Main article: Transmission electron microscope

The original form of the electron microscope, the transmission electron microscope (TEM), uses a high voltage electron beam to illuminate the specimen and create an image. An electron beam is produced by an electron gun, with the electrons typically at 40 to 400 keV, focused by electromagnetic lenses, and transmitted through the specimen. When it emerges from the specimen, the electron beam carries information about the structure of the specimen that is magnified by lenses of the microscope. The spatial variation in this information (the "image") may be viewed by projecting the magnified electron image onto a detector. For example, the image may be viewed directly by an operator using a fluorescent viewing screen coated with a phosphor or scintillator material such as zinc sulfide. A high-resolution phosphor may also be coupled by means of a lens optical system or a fibre optic light-guide to the sensor of a digital camera. Direct electron detectors have no scintillator and are directly exposed to the electron beam, which addresses some of the limitations of scintillator-coupled cameras.

The resolution of TEMs is limited primarily by spherical aberration, but a new generation of hardware correctors can reduce spherical aberration to increase the resolution in high-resolution transmission electron microscopy (HRTEM) to below 0.5 angstrom (50 picometres), enabling magnifications above 50 million times. The ability of HRTEM to determine the positions of atoms within materials is useful for nano-technologies research and development.

Main article: Electron diffraction

Transmission electron microscopes are often used in electron diffraction mode. The advantages of electron diffraction over X-ray crystallography are that the specimen need not be a single crystal or even a polycrystalline powder.

Scanning transmission electron microscope (STEM)

Main article: Scanning transmission electron microscopy

The STEM rasters a focused incident probe across a specimem. The high resolution of the TEM is thus possible in STEM. The focusing action (and aberrations) occur before the electrons hit the specimen in the STEM, but afterward in the TEM. The STEMs use of SEM-like beam rastering simplifies annular dark-field imaging, and other analytical techniques, but also means that image data is acquired in serial rather than in parallel fashion.

Scanning electron microscope (SEM)

Main article: Scanning electron microscope

Image of Bacillus subtilis taken with a 1960s electron microscope

The SEM produces images by probing the specimen with a focused electron beam that is scanned across the specimen (raster scanning). When the electron beam interacts with the specimen, it loses energy by a variety of mechanisms. The lost energy is converted into alternative forms such as heat, emission of low-energy secondary electrons and high-energy backscattered electrons, light emission (cathodoluminescence) or X-ray emission, all of which provide signals carrying information about the properties of the specimen surface, such as its topography and composition. The image displayed by an SEM maps the varying intensity of any of these signals into the image in a position corresponding to the position of the beam on the specimen when the signal was generated. In the SEM image of an ant shown, the image was constructed from signals produced by a secondary electron detector, the normal or conventional imaging mode in most SEMs.

Generally, the image resolution of an SEM is lower than that of a TEM. However, because the SEM images the surface of a sample rather than its interior, the electrons do not have to travel through the sample. This reduces the need for extensive sample preparation to thin the specimen to electron transparency. The SEM also has a great depth of field, and so can produce images that are good representations of the three-dimensional surface shape of the sample.

In their most common configurations, electron microscopes produce images with a single brightness value per pixel, with the results usually rendered in greyscale. However, often these images are then colourized through the use of feature-detection software, or simply by hand-editing using a graphics editor. This may be done to clarify structure or for aesthetic effect and generally does not add new information about the specimen.

Sample preparation for TEM

An insect coated in gold for viewing with a scanning electron microscope

See also: TEM Sample preparation, Ultramicrotomy, Staining, Cryofixation, Chemical milling, and Sputtering

Materials to be viewed in a transmission electron microscope may require processing to produce a suitable sample. The technique required varies depending on the specimen and the analysis required:

• Chemical fixation – for biological specimens this aims to stabilize the specimen's mobile macromolecular structure by chemical crosslinking of proteins with aldehydes such as formaldehyde and glutaraldehyde, and lipids with osmium tetroxide.
• Cryofixation – freezing a specimen so that the water forms vitreous (non-crystalline) ice. This preserves the specimen in a snapshot of its native state. Methods to achieve this vitrification include plunge freezing rapidly in liquid ethane, and high pressure freezing. An entire field called cryo-electron microscopy has branched from this technique. With the development of cryo-electron microscopy of vitreous sections (CEMOVIS), it is now possible to observe samples from virtually any biological specimen close to its native state.
• Dehydration – replacement of water with organic solvents such as ethanol or acetone, followed by critical point drying or infiltration with embedding resins. See also freeze drying.
• Embedding, biological specimens – after dehydration, tissue for observation in the transmission electron microscope is embedded so it can be sectioned ready for viewing. To do this the tissue is passed through a 'transition solvent' such as propylene oxide (epoxypropane) or acetone and then infiltrated with an epoxy resin such as Araldite, Epon, or Durcupan; tissues may also be embedded directly in water-miscible acrylic resin. After the resin has been polymerized (hardened) the sample is thin sectioned (ultrathin sections) and stained.
• Embedding, materials – after embedding in resin, the specimen is usually ground and polished to a mirror-like finish using ultra-fine abrasives.
• Freeze-fracture or freeze-etch – a preparation method particularly useful for examining lipid membranes and their incorporated proteins in "face on" view.

  

  Freeze-fracturing helps to peel open membranes to allow visualization of what is inside

  

  External face of bakers yeast membrane showing the small holes where proteins are fractured out, sometimes as small ring patterns.

  The fresh tissue or cell suspension is frozen rapidly (cryofixation), then fractured by breaking (or by using a microtome) while maintained at liquid nitrogen temperature. The cold fractured surface (sometimes "etched" by increasing the temperature to about −100 °C for several minutes to let some ice sublime) is then shadowed with evaporated platinum or gold at an average angle of 45° in a high vacuum evaporator. The second coat of carbon, evaporated perpendicular to the average surface plane is often performed to improve the stability of the replica coating. The specimen is returned to room temperature and pressure, then the extremely fragile "pre-shadowed" metal replica of the fracture surface is released from the underlying biological material by careful chemical digestion with acids, hypochlorite solution or SDS detergent. The still-floating replica is thoroughly washed free from residual chemicals, carefully fished up on fine grids, dried then viewed in the TEM.
• Freeze-fracture replica immunogold labeling (FRIL) – the freeze-fracture method has been modified to allow the identification of the components of the fracture face by immunogold labeling. Instead of removing all the underlying tissue of the thawed replica as the final step before viewing in the microscope the tissue thickness is minimized during or after the fracture process. The thin layer of tissue remains bound to the metal replica so it can be immunogold labeled with antibodies to the structures of choice. The thin layer of the original specimen on the replica with gold attached allows the identification of structures in the fracture plane. There are also related methods which label the surface of etched cells and other replica labeling variations.
• Ion beam milling – thins samples until they are transparent to electrons by firing ions (typically argon) at the surface from an angle and sputtering material from the surface. A subclass of this is focused ion beam milling, where gallium ions are used to produce an electron transparent membrane or 'lamella' in a specific region of the sample, for example through a device within a microprocessor or a focused ion beam SEM. Ion beam milling may also be used for cross-section polishing prior to analysis of materials that are difficult to prepare using mechanical polishing.
• Negative stain – suspensions containing nanoparticles or fine biological material (such as viruses and bacteria) are briefly mixed with a dilute solution of an electron-opaque solution such as ammonium molybdate, uranyl acetate (or formate), or phosphotungstic acid. This mixture is applied to a suitably coated EM grid, blotted, then allowed to dry. Viewing of this preparation in the TEM should be carried out without delay for best results. The method is important in microbiology for fast but crude morphological identification, but can also be used as the basis for high-resolution 3D reconstruction using EM tomography methodology when carbon films are used for support. Negative staining is also used for observation of nanoparticles.
• Sectioning – produces thin slices of the specimen, semitransparent to electrons. These can be cut using ultramicrotomy on an ultramicrotome with a glass or diamond knife to produce ultra-thin sections about 60–90 nm thick. Disposable glass knives are also used because they can be made in the lab and are much cheaper. Sections can also be created in situ by milling in a focused ion beam SEM, where the section is known as a lamella.
• Staining – uses heavy metals such as lead, uranium or tungsten to scatter imaging electrons and thus give contrast between different structures, since many (especially biological) materials are nearly "transparent" to electrons (weak phase objects). In biology, specimens can be stained "en bloc" before embedding and also later after sectioning. Typically thin sections are stained for several minutes with an aqueous or alcoholic solution of uranyl acetate followed by aqueous lead citrate.

Disadvantages

JEOL transmission and scanning electron microscope made in the mid-1970s

Electron microscopes are expensive to build and maintain. Microscopes designed to achieve high resolutions must be housed in stable buildings (sometimes underground) with special services such as magnetic field canceling systems.

The samples largely have to be viewed in vacuum, as the molecules that make up air would scatter the electrons. An exception is liquid-phase electron microscopy using either a closed liquid cell or an environmental chamber, for example, in the environmental scanning electron microscope, which allows hydrated samples to be viewed in a low-pressure (up to 20 Torr or 2.7 kPa) wet environment. Various techniques for in situ electron microscopy of gaseous samples have been developed.

Scanning electron microscopes operating in conventional high-vacuum mode usually image conductive specimens; therefore non-conductive materials require conductive coating (gold/palladium alloy, carbon, osmium, etc.). The low-voltage mode of modern microscopes makes possible the observation of non-conductive specimens without coating. Non-conductive materials can be imaged also by a variable pressure (or environmental) scanning electron microscope.

Small, stable specimens such as carbon nanotubes, diatom frustules and small mineral crystals (asbestos fibres, for example) require no special treatment before being examined in the electron microscope. Samples of hydrated materials, including almost all biological specimens, have to be prepared in various ways to stabilize them, reduce their thickness (ultrathin sectioning) and increase their electron optical contrast (staining). These processes may result in artifacts, but these can usually be identified by comparing the results obtained by using radically different specimen preparation methods. Since the 1980s, analysis of cryofixed, vitrified specimens has also become increasingly used by scientists, further confirming the validity of this technique.