Given 7 numbers, two 1’s and five 3’s, how many different ways can these numbers be arranged?

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Mathematical version of an order change

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.[1]

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.[2][3] 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 α {\displaystyle \alpha }

α ( 1 ) = 3 , α ( 2 ) = 1 , α ( 3 ) = 2 {\displaystyle \alpha (1)=3,\quad \alpha (2)=1,\quad \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 { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}}

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.

History

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

Al-Khalil (717–786), an Arab mathematician and cryptographer, wrote the Book of Cryptographic Messages. It contains the first use of permutations and combinations, to list all possible Arabic words with and without vowels.[4]

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

The product of multiplication of the arithmetical series beginning and increasing by unity and continued to the number of places, will be the variations of number with specific figures.[5]

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.[6] 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".[7] 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.[8] 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;[9]

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.[10]

A first case in which seemingly unrelated mathematical questions were studied with the help of permutations occurred around 1770, when Joseph Louis Lagrange, in the study of polynomial equations, observed that properties of the permutations of the roots of an equation are related to the possibilities to solve it. This line of work ultimately resulted, through the work of Évariste Galois, in Galois theory, which gives a complete description of what is possible and impossible with respect to solving polynomial equations (in one unknown) by radicals. In modern mathematics, there are many similar situations in which understanding a problem requires studying certain permutations related to it.

Permutations without repetitions

The simplest example of permutations is permutations without repetitions where we consider the number of possible ways of arranging 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 3 ! = 1 ⋅ 2 ⋅ 3 = 6 {\displaystyle 3!=1\cdot 2\cdot 3=6}

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 n P k {\displaystyle nPk}

n ( n − 1 ) ⋯ ( n − k + 1 ) {\displaystyle n(n-1)\cdots (n-k+1)}

n ! / ( n − k ) ! {\displaystyle n!/(n-k)!}

).

Definition

In mathematics texts it is customary to denote permutations using lowercase Greek letters. Commonly, either α {\displaystyle \alpha } and β {\displaystyle \beta }

σ , τ {\displaystyle \sigma ,\tau }

π {\displaystyle \pi }

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 S n {\displaystyle S_{n}}

σ {\displaystyle \sigma }

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

In general, composition of two permutations is not commutative, that is, π σ ≠ σ π . {\displaystyle \pi \sigma \neq \sigma \pi .}

As a bijection from a set to itself, a permutation is a function that performs a rearrangement of a set, and is not an arrangement itself. An older and more elementary viewpoint is that permutations are the arrangements themselves. To distinguish between these two, the identifiers active and passive are sometimes prefixed to the term permutation, whereas in older terminology substitutions and permutations are used.[14]

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 σ {\displaystyle \sigma } defined by σ ( 7 ) = 7 {\displaystyle \sigma (7)=7}

( 7 ) {\displaystyle (\,7\,)}

π ( 2 ) = 3 {\displaystyle \pi (2)=3}

π ( 3 ) = 2 {\displaystyle \pi (3)=2}

( 2 3 ) {\displaystyle (\,2\,3\,)}

An element in a 1-cycle ( x ) {\displaystyle (\,x\,)}

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,[15] 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

σ = ( 1 2 3 4 5 2 5 4 3 1 ) ; {\displaystyle \sigma ={\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}};}

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

σ = ( 3 2 5 1 4 4 5 1 2 3 ) , {\displaystyle \sigma ={\begin{pmatrix}3&2&5&1&4\\4&5&1&2&3\end{pmatrix}},}

or

σ = ( 5 1 4 3 2 1 2 3 4 5 ) . {\displaystyle \sigma ={\begin{pmatrix}5&1&4&3&2\\1&2&3&4&5\end{pmatrix}}.}

One-line notation

If there is a "natural" order for the elements of S,[a] say x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}}

σ = ( x 1 x 2 x 3 ⋯ x n σ ( x 1 ) σ ( x 2 ) σ ( x 3 ) ⋯ σ ( x n ) ) . {\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

( σ ( x 1 ) σ ( x 2 ) σ ( x 3 ) ⋯ σ ( x n ) ) {\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.[16][17] 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.[18] 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 σ {\displaystyle \sigma } in cycle notation, one proceeds as follows:

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

( 1 2 5 ) ( 3 4 ) = ( 3 4 ) ( 1 2 5 ) . {\displaystyle (\,1\,2\,5\,)(\,3\,4\,)=(\,3\,4\,)(\,1\,2\,5\,).}

( 1 2 5 ) ( 3 4 ) = ( 5 1 2 ) ( 3 4 ) = ( 2 5 1 ) ( 4 3 ) . {\displaystyle (\,1\,2\,5\,)(\,3\,4\,)=(\,5\,1\,2\,)(\,3\,4\,)=(\,2\,5\,1\,)(\,4\,3\,).}

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 σ ( x ) = x {\displaystyle \sigma (x)=x}

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,

[ ( 1 2 5 ) ( 3 4 ) ] − 1 = ( 5 2 1 ) ( 4 3 ) . {\displaystyle [(\,1\,2\,5\,)(\,3\,4\,)]^{-1}=(\,5\,2\,1\,)(\,4\,3\,).}

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.[22] The canonical cycle notation does not omit one-cycles.

Richard P. Stanley calls the same choice of representation the "standard representation" of a permutation,[23] and Martin Aigner uses the term "standard form" for the same notion.[18] 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.[24]

Composition of permutations

There are two ways to denote the composition of two permutations. σ ⋅ π {\displaystyle \sigma \cdot \pi }

σ ( π ( x ) ) {\displaystyle \sigma (\pi (x))}

Since function composition is associative, so is the composition operation on permutations: ( σ π ) τ = σ ( π τ ) {\displaystyle (\sigma \pi )\tau =\sigma (\pi \tau )}

Some authors prefer the leftmost factor acting first,[26][27][28] 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 xσ; then the product is defined by xσ·π = (xσ)π. However this gives a different rule for multiplying permutations; this article uses the definition where the rightmost permutation is applied first.

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[c]). These objects are also known as partial permutations or as sequences without repetition, terms that avoid confusion with the other, more common, meaning of "permutation". The number of such k {\displaystyle k}

n {\displaystyle n}

P k n {\displaystyle P_{k}^{n}}

n P k {\displaystyle _{n}P_{k}}

n P k {\displaystyle ^{n}P_{k}}

P n , k {\displaystyle P_{n,k}}

P ( n , k ) {\displaystyle P(n,k)}

P ( n , k ) = n ⋅ ( n − 1 ) ⋅ ( n − 2 ) ⋯ ( n − k + 1 ) ⏟ k   f a c t o r s {\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

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

The product is well defined without the assumption that n {\displaystyle n} is a non-negative integer, and is of importance outside combinatorics as well; it is known as the Pochhammer symbol ( n ) k {\displaystyle (n)_{k}}

n k _ {\displaystyle n^{\underline {k}}}

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:

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

These numbers are also known as binomial coefficients and are denoted by ( n k ) {\displaystyle {\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 general. They are also called words over the alphabet S in some contexts. If the set S has n elements, the number of k-tuples over S is n k . {\displaystyle n^{k}.}

Permutations of multisets

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

m 2 {\displaystyle m_{2}}

m l {\displaystyle m_{l}}

( n m 1 , m 2 , … , m l ) = n ! m 1 ! m 2 ! ⋯ m l ! = ( ∑ i = 1 l m i ) ! ∏ i = 1 l m i ! . {\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:[31]

11 ! 1 ! 4 ! 4 ! 2 ! = 34650 {\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.[32][e] 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.

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.

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).[33]

Cycle type

The cycles (including the fixed points) of a permutation σ {\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 σ {\displaystyle \sigma } . There is a "1" in the cycle type for every fixed point of σ {\displaystyle \sigma } , a "2" for every transposition, and so on. The cycle type of β = ( 1 2 5 ) ( 3 4 ) ( 6 8 ) ( 7 ) {\displaystyle \beta =(1\,2\,5\,)(\,3\,4\,)(6\,8\,)(\,7\,)}

( 3 , 2 , 2 , 1 ) . {\displaystyle (3,2,2,1).}

This may also be written in a more compact form as [112231]. More precisely, the general form is [ 1 α 1 2 α 2 ⋯ n α n ] {\displaystyle [1^{\alpha _{1}}2^{\alpha _{2}}\dotsm n^{\alpha _{n}}]}

α 1 , … , α n {\displaystyle \alpha _{1},\ldots ,\alpha _{n}}

n ! 1 α 1 2 α 2 ⋯ n α n α 1 ! α 2 ! ⋯ α n ! {\displaystyle {\frac {n!}{1^{\alpha _{1}}2^{\alpha _{2}}\dotsm n^{\alpha _{n}}\alpha _{1}!\alpha _{2}!\dotsm \alpha _{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 σ {\displaystyle \sigma } by another permutation π {\displaystyle \pi } , which means forming the product π σ π − 1 {\displaystyle \pi \sigma \pi ^{-1}}

Permutation order

The order of a permutation σ {\displaystyle \sigma } is the smallest positive integer m so that σ m = i d {\displaystyle \sigma ^{m}=\mathrm {id} }

( 1 3 2 ) ( 4 5 ) {\displaystyle (\,1\,3\,2)(\,4\,5\,)}

2 ⋅ 3 = 6 {\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.[36] 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 sgn ⁡ σ {\displaystyle \operatorname {sgn} \sigma }

sgn ⁡ σ = + 1 {\displaystyle \operatorname {sgn} \sigma =+1}

sgn ⁡ σ = − 1 {\displaystyle \operatorname {sgn} \sigma =-1}

sgn ⁡ ( σ π ) = sgn ⁡ σ ⋅ sgn ⁡ π . {\displaystyle \operatorname {sgn} (\sigma \pi )=\operatorname {sgn} \sigma \cdot \operatorname {sgn} \pi .}

It follows that sgn ⁡ ( σ σ − 1 ) = + 1. {\displaystyle \operatorname {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 M σ {\displaystyle M_{\sigma }}

M σ M π = M σ ∘ π {\displaystyle M_{\sigma }M_{\pi }=M_{\sigma \circ \pi }}

e i {\displaystyle {\bf {e}}_{i}}

n × 1 {\displaystyle n\times 1}

M σ e i = e σ ( i ) {\displaystyle M_{\sigma }{\bf {e}}_{i}={\bf {e}}_{\sigma (i)}}

For example, with this convention, the matrix associated to the permutation σ ( 1 , 2 , 3 ) = ( 2 , 1 , 3 ) {\displaystyle \sigma (1,2,3)=(2,1,3)}

( 0 1 0 1 0 0 0 0 1 ) {\displaystyle {\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}}}

π ( 1 , 2 , 3 ) = ( 2 , 3 , 1 ) {\displaystyle \pi (1,2,3)=(2,3,1)}

( 0 0 1 1 0 0 0 1 0 ) {\displaystyle {\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}}

( σ ∘ π ) ( 1 , 2 , 3 ) = σ ( 2 , 3 , 1 ) = ( 1 , 3 , 2 ) {\displaystyle (\sigma \circ \pi )(1,2,3)=\sigma (2,3,1)=(1,3,2)}

M ( 2 , 1 , 3 ) M ( 2 , 3 , 1 ) = ( 0 1 0 1 0 0 0 0 1 ) ( 0 0 1 1 0 0 0 1 0 ) = ( 1 0 0 0 0 1 0 1 0 ) = M ( 1 , 3 , 2 ) . {\displaystyle M_{(2,1,3)}M_{(2,3,1)}={\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_{(1,3,2)}.}

It is also common in the literature to find the inverse convention, where a permutation σ is associated to the matrix P σ = ( M σ ) − 1 = ( M σ ) T {\displaystyle P_{\sigma }=(M_{\sigma })^{-1}=(M_{\sigma })^{T}}

P σ P π = P π ∘ σ {\displaystyle P_{\sigma }P_{\pi }=P_{\pi \circ \sigma }}

1 × n {\displaystyle 1\times n}

( e i ) T {\displaystyle ({\bf {e}}_{i})^{T}}

( e i ) T P σ = ( e σ ( 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.

Permutations of totally ordered sets

In some applications, the elements of the set being permuted will be compared with each other. This requires that the set S has a total order so that any two elements can be compared. The set {1, 2, ..., n} is totally ordered by the usual "≤" relation and so it is the most frequently used set in these applications, but in general, any totally ordered set will do. In these applications, the ordered arrangement view of a permutation is needed to talk about the positions in a permutation.

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

Ascents, descents, runs and excedances

An ascent of a permutation σ of n is any position i < n where the following value is bigger than the current one. That is, if σ = σ1σ2...σn, then i is an ascent if σi < σi+1.

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

Similarly, a descent is a position i < n with σi > σi+1, so every i with 1 ≤ i < n {\displaystyle 1\leq i<n}

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.[37]

The number of permutations of n with k ascents is (by definition) the Eulerian number ⟨ n k ⟩ {\displaystyle \textstyle \left\langle {n \atop k}\right\rangle }

An excedance of a permutation σ1σ2...σn is an index j such that σj > j. If the inequality is not strict (that is, σj ≥ j), then j is called a weak excedance. The number of n-permutations with k excedances coincides with the number of n-permutations with k descents.[39]

Foata's transition lemma

There is a relationship between the one-line notation and the canonical cycle notation. Consider the permutation ( 2 ) ( 3 1 ) {\displaystyle (\,2\,)(\,3\,1\,)}

231 {\displaystyle 231}

Let f ( p ) = q {\displaystyle f(p)=q}

q {\displaystyle q}

p {\displaystyle p}

f {\displaystyle f}

f − 1 ( q ) = p {\displaystyle f^{-1}(q)=p}

q = q 1 q 2 ⋯ q n {\displaystyle q=q_{1}q_{2}\cdots q_{n}}

q 1 {\displaystyle q_{1}}

j {\displaystyle j}

q j > q 1 {\displaystyle q_{j}>q_{1}}

q j {\displaystyle q_{j}}

For example, in the permutation q = 312548976 {\displaystyle q=312548976}

( 3 1 2 ) {\displaystyle (\,3\,1\,2\,)}

( 5 4 ) {\displaystyle (\,5\,4\,)}

( 8 ) {\displaystyle (\,8\,)}

( 9 7 6 ) {\displaystyle (\,9\,7\,6\,)}

p = ( 3 1 2 ) ( 5 4 ) ( 8 ) ( 9 7 6 ) {\displaystyle p=(\,3\,1\,2\,)(\,5\,4\,)(\,8\,)(\,9\,7\,6\,)}

The following table shows both q {\displaystyle q} and p {\displaystyle p} for the six permutations of 123 {\displaystyle 123}

q = f ( p ) {\displaystyle q=f(p)}	p = f − 1 ( q ) {\displaystyle p=f^{-1}(q)}
123 = ( 1 ) ( 2 ) ( 3 ) {\displaystyle \mathbf {123} =(\,1\,)(\,2\,)(\,3\,)}	123 = ( 1 ) ( 2 ) ( 3 ) {\displaystyle 123=\mathbf {(\,1\,)(\,2\,)(\,3\,)} }
132 = ( 1 ) ( 3 2 ) {\displaystyle \mathbf {132} =(\,1\,)(\,3\,2\,)}	132 = ( 1 ) ( 3 2 ) {\displaystyle 132=\mathbf {(\,1\,)(\,3\,2\,)} }
213 = ( 2 1 ) ( 3 ) {\displaystyle \mathbf {213} =(\,2\,1\,)(\,3\,)}	213 = ( 2 1 ) ( 3 ) {\displaystyle 213=\mathbf {(\,2\,1\,)(\,3\,)} }
231 = ( 3 1 2 ) {\displaystyle \mathbf {231} =(\,3\,1\,2\,)}	321 = ( 2 ) ( 3 1 ) {\displaystyle 321=\mathbf {(\,2\,)(\,3\,1\,)} }
312 = ( 3 2 1 ) {\displaystyle \mathbf {312} =(\,3\,2\,1\,)}	231 = ( 3 1 2 ) {\displaystyle 231=\mathbf {(\,3\,1\,2\,)} }
321 = ( 2 ) ( 3 1 ) {\displaystyle \mathbf {321} =(\,2\,)(\,3\,1\,)}	312 = ( 3 2 1 ) {\displaystyle 312=\mathbf {(\,3\,2\,1\,)} }

As a first corollary, the number of n-permutations with exactly k left-to-right maxima is also equal to the signless Stirling number of the first kind, c ( n , k ) {\displaystyle c(n,k)}

Inversions

Main article: Inversion (discrete mathematics)

An inversion of a permutation σ is a pair (i, j) of positions where the entries of a permutation are in the opposite order: i < j {\displaystyle i<j}

σ i > σ j {\displaystyle \sigma _{i}>\sigma _{j}}

Sometimes an inversion is defined as the pair of values (σi,σj) 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,[43] it is the coefficient of Xk in the expansion of the product

∏ m = 1 n ∑ i = 0 m − 1 X i = 1 ( 1 + X ) ( 1 + X + X 2 ) ⋯ ( 1 + X + X 2 + ⋯ + X n − 1 ) , {\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),}

Let σ ∈ S n , i , j ∈ { 1 , 2 , … , n } {\displaystyle \sigma \in S_{n},i,j\in \{1,2,\dots ,n\}}

σ ( i ) > σ ( j ) {\displaystyle \sigma (i)>\sigma (j)}

( i , j ) {\displaystyle (i,j)}

σ ( i ) − σ ( j ) {\displaystyle \sigma (i)-\sigma (j)}

∑ i < j , σ ( i ) > σ ( j ) ( σ ( i ) − σ ( j ) ) = | { τ ∈ S n ∣ τ ≤ σ , τ  is bigrassmannian } {\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 }

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 dn, dn−1, ..., d2, d1, where di is a non-negative integer less than i (one may omit d1, as it is always 0, but its presence makes the subsequent conversion to a permutation easier to describe). The first step then is 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.

σ = ( 6 , 3 , 8 , 1 , 4 , 9 , 7 , 2 , 5 ) {\displaystyle \sigma =(6,3,8,1,4,9,7,2,5)}

In the Lehmer code for a permutation σ, the number dn represents the choice made for the first term σ1, the number dn−1 represents the choice made for the second term σ2 among the remaining n − 1 elements of the set, and so forth. More precisely, each dn+1−i gives the number of remaining elements strictly less than the term σi. Since those remaining elements are bound to turn up as some later term σj, the digit dn+1−i counts the inversions (i,j) involving i as smaller index (the number of values j for which i < j and σi > σj). The inversion table for σ is quite similar, but here dn+1−k counts the number of inversions (i,j) where k = σj occurs as the smaller of the two values appearing in inverted order.[44] Both encodings can be visualized by an n by n Rothe diagram[45] (named after Heinrich August Rothe) in which dots at (i,σi) mark the entries of the permutation, and a cross at (i,σj) marks the inversion (i,j); by the definition of inversions a cross appears in any square that comes both before the dot (j,σj) in its column, and before the dot (i,σi) in its row. The Lehmer code lists the numbers of crosses in successive rows, while the inversion table lists the numbers of crosses in successive columns; it is just the Lehmer code for the inverse permutation, and vice versa.

To effectively convert a Lehmer code dn, dn−1, ..., d2, d1 into a permutation of an ordered set S, one can start with a list of the elements of S in increasing order, and for i increasing from 1 to n set σi to the element in the list that is preceded by dn+1−i other ones, and remove that element from the list. To convert an inversion table dn, dn−1, ..., d2, d1 into the corresponding permutation, one can traverse the numbers from d1 to dn while inserting the elements of S from largest to smallest into an initially empty sequence; at the step using the number d from the inversion table, the element from S inserted into the sequence at the point where it is preceded by d elements already present. Alternatively one could process the numbers from the inversion table and the elements of S both in the opposite order, starting with a row of n empty slots, and at each step place the element from S into the empty slot that is preceded by d other empty slots.

Converting successive natural numbers to the factorial number system produces those sequences in lexicographic order (as is the case with any mixed radix number system), and further converting them to permutations preserves the lexicographic ordering, provided the Lehmer code interpretation is used (using inversion tables, one gets a different ordering, where one starts by comparing permutations by the place of their entries 1 rather than by the value of their first entries). The sum of the numbers in the factorial number system representation gives the number of inversions of the permutation, and the parity of that sum gives the signature of the permutation. Moreover, the positions of the zeroes in the inversion table give the values of left-to-right maxima of the permutation (in the example 6, 8, 9) while the positions of the zeroes in the Lehmer code are the positions of the right-to-left minima (in the example positions the 4, 8, 9 of the values 1, 2, 5); this allows computing the distribution of such extrema among all permutations. A permutation with Lehmer code dn, dn−1, ..., d2, d1 has an ascent n − i if and only if di ≥ di+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,...,dn satisfying 0 ≤ di < i (since d1 is always zero it may be omitted) and to convert it to a permutation through a bijective correspondence. For the latter correspondence one could interpret the (reverse) sequence as a Lehmer code, and this gives a generation method first published in 1938 by Ronald Fisher and Frank Yates.[46] While at the time computer implementation was not an issue, this method suffers from the difficulty sketched above to convert from Lehmer code to permutation efficiently. This can be remedied by using a different bijective correspondence: after using di to select an element among i remaining elements of the sequence (for decreasing values of i), rather than removing the element and compacting the sequence by shifting down further elements one place, one swaps the element with the final remaining element. Thus the elements remaining for selection form a consecutive range at each point in time, even though they may not occur in the same order as they did in the original sequence. The mapping from sequence of integers to permutations is somewhat complicated, but it can be seen to produce each permutation in exactly one way, by an immediate induction. When the selected element happens to be the final remaining element, the swap operation can be omitted. This does not occur sufficiently often to warrant testing for the condition, but the final element must be included among the candidates of the selection, to guarantee that all permutations can be generated.

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

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

If di+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.[47]

Generation in lexicographic order

There are many ways to systematically generate all permutations of a given sequence.[48] 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.[49]

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.[50]

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.[49]

An alternative to Steinhaus–Johnson–Trotter is Heap's algorithm,[51] said by Robert Sedgewick in 1977 to be the fastest algorithm of generating permutations in applications.[48]

The following figure shows the output of all three aforementioned algorithms for generating all permutations of length n = 4 {\displaystyle n=4}

1. Lexicographic ordering;
2. Steinhaus–Johnson–Trotter algorithm;
3. Heap's algorithm;
4. Ehrlich's star-transposition algorithm:[49] in each step, the first entry of the permutation is exchanged with a later entry;
5. Zaks' prefix reversal algorithm:[53] in each step, a prefix of the current permutation is reversed to obtain the next permutation;
6. Sawada-Williams' algorithm:[54] 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:[55] each permutation differs from the previous one by a cyclic left-shift of some prefix by one position;
8. Single-track ordering:[56] each column is a cyclic shift of the other columns;
9. Single-track Gray code:[56] 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.[57][unreliable source?] Generation of these alternate permutations is needed before they are analyzed to determine if they are meandric or not.

The algorithm is recursive. The following table exhibits a step in the procedure. In the previous step, all alternate permutations of length 5 have been generated. Three copies of each of these have a "6" added to the right end, and then a different transposition involving this last entry and a previous entry in an even position is applied (including the identity; that is, no transposition).

Applications

Permutations are used in the interleaver component of the error detection and correction algorithms, such as turbo codes, for example 3GPP Long Term Evolution mobile telecommunication standard uses these ideas (see 3GPP technical specification 36.212[58]). 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.[59]

See also

• 
  Mathematics portal

• Alternating permutation
• Convolution
• Cyclic order
• Even and odd permutations
• Josephus permutation
• Levi-Civita symbol
• List of permutation topics
• Major index
• Permutation category
• Permutation group
• Permutation pattern
• Permutation representation (symmetric group)
• Probability
• Rencontres numbers
• Sorting network
• Substitution cipher
• Superpattern
• Superpermutation
• Twelvefold way
• Weak order of permutations

Notes

1. ^ The order is often implicitly understood. A set of integers is naturally written from smallest to largest; a set of letters is written in lexicographic order. For other sets, a natural order needs to be specified explicitly.
2. ^ 1 is frequently used to represent the identity element in a non-commutative group
3. ^ More precisely, variations without repetition. The term is still common in other languages and appears in modern English most often in translation.
4. ^ The natural order in this example is the order of the letters in the original word.
5. ^ In older texts circular permutation was sometimes used as a synonym for cyclic permutation, but this is no longer done. See Carmichael (1956, p. 7)

References

1. ^ Webster (1969)
2. ^ McCoy (1968, p. 152)
3. ^ Nering (1970, p. 86)
4. ^ Broemeling, Lyle D. (1 November 2011). "An Account of Early Statistical Inference in Arab Cryptology". The American Statistician. 65 (4): 255–257. doi:10.1198/tas.2011.10191. S2CID 123537702.
5. ^ Biggs, N. L. (1979). "The Roots of Combinatorics". Historia Math. 6 (2): 109–136. doi:10.1016/0315-0860(79)90074-0.
6. ^ Stedman 1677, p. 4.
7. ^ Stedman 1677, p. 5.
8. ^ Stedman 1677, pp. 6–7.
9. ^ Stedman 1677, p. 8.
10. ^ Stedman 1677, pp. 13–18.
11. ^ "Combinations and Permutations". www.mathsisfun.com. Retrieved 2020-09-10.
12. ^ Weisstein, Eric W. "Permutation". mathworld.wolfram.com. Retrieved 2020-09-10.
13. ^ Scheinerman, Edward A. (March 5, 2012). "Chapter 5: Functions". Mathematics: A Discrete Introduction (3rd ed.). Cengage Learning. p. 188. ISBN 978-0840049421. Archived from the original on February 5, 2020. Retrieved February 5, 2020. It is customary to use lowercase Greek letters (especially π, σ, and τ) to stand for permutations.
14. ^ Cameron 1994, p. 29, footnote 3.
15. ^ Wussing, Hans (2007), The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, Courier Dover Publications, p. 94, ISBN 9780486458687, Cauchy used his permutation notation—in which the arrangements are written one below the other and both are enclosed in parentheses—for the first time in 1815.
16. ^ Bogart 1990, p. 17
17. ^ Gerstein 1987, p. 217
18. ^ a b Aigner, Martin (2007). A Course in Enumeration. Springer GTM 238. pp. 24–25. ISBN 978-3-540-39035-0.
19. ^ Hall 1959, p. 54
20. ^ Rotman 2002, p. 41
21. ^ Bogart 1990, p. 487
22. ^ Bona 2012, p.87 [Note that the book has a typo/error here, as it gives (45) instead of (54).]
23. ^ a b Stanley, Richard P. (2012). Enumerative Combinatorics: Volume I, Second Edition. Cambridge University Press. p. 23. ISBN 978-1-107-01542-5.
24. ^ Kitaev, Sergey (2011). Patterns in Permutations and Words. Springer Science & Business Media. p. 119. ISBN 978-3-642-17333-2.
25. ^ Biggs, Norman L.; White, A. T. (1979). Permutation groups and combinatorial structures. Cambridge University Press. ISBN 978-0-521-22287-7.
26. ^ Dixon, John D.; Mortimer, Brian (1996). Permutation Groups. Springer. ISBN 978-0-387-94599-6.
27. ^ Cameron, Peter J. (1999). Permutation groups. Cambridge University Press. ISBN 978-0-521-65302-2.
28. ^ Jerrum, M. (1986). "A compact representation of permutation groups". J. Algorithms. 7 (1): 60–78. doi:10.1016/0196-6774(86)90038-6. S2CID 18896625.
29. ^ Charalambides, Ch A. (2002). Enumerative Combinatorics. CRC Press. p. 42. ISBN 978-1-58488-290-9.
30. ^ Brualdi 2010, p. 46, Theorem 2.4.2
31. ^ Brualdi 2010, p. 47
32. ^ Brualdi 2010, p. 39
33. ^ Bona 2012, pp. 97–103.
34. ^ Sagan, Bruce (2001), The Symmetric Group (2 ed.), Springer, p. 3
35. ^ Humphreys 1996, p. 84.
36. ^ Hall 1959, p. 60
37. ^ Bóna 2004, p. 4f.
38. ^ Bona 2012, pp. 4–5.
39. ^ Bona 2012, p. 25.
40. ^ a b c Bona 2012, pp. 109–110.
41. ^ Slocum, Jerry; Weisstein, Eric W. (1999). "15 – puzzle". MathWorld. Wolfram Research, Inc. Retrieved October 4, 2014.
42. ^ Bóna 2004, p. 43.
43. ^ Bóna 2004, pp. 43ff.
44. ^ Knuth 1973, p. 12.
45. ^ H. A. Rothe, Sammlung combinatorisch-analytischer Abhandlungen 2 (Leipzig, 1800), 263–305. Cited in Knuth 1973, p. 14
46. ^ Fisher, R.A.; Yates, F. (1948) [1938]. Statistical tables for biological, agricultural and medical research (3rd ed.). London: Oliver & Boyd. pp. 26–27. OCLC 14222135.
47. ^ Bacher, A.; Bodini, O.; Hwang, H.K.; Tsai, T.H. (2017). "Generating Random Permutations by Coin Tossing: Classical Algorithms, New Analysis, and Modern Implementation" (ACM Trans. Algorithms 13(2): 24:1–24:43 ed.). pp. 24–43.
48. ^ a b Sedgewick, R (1977). "Permutation generation methods" (PDF). Computing Surveys. 9 (2): 137–164. doi:10.1145/356689.356692. S2CID 12139332.
49. ^ a b c Knuth 2005, pp. 1–26.
50. ^ "std::next_permutation". cppreference.com. 4 December 2017. Retrieved 31 March 2018.
51. ^ Heap, B. R. (1963). "Permutations by Interchanges". The Computer Journal. 6 (3): 293–298. doi:10.1093/comjnl/6.3.293.
52. ^ Mütze, Torsten; Sawada, Joe; Williams, Aaron. "Generate permutations". Combinatorial Object Server. Retrieved May 29, 2019.
53. ^ Zaks, S. (1984). "A new algorithm for generation of permutations". BIT Numerical Mathematics. 24 (2): 196–204. doi:10.1007/BF01937486. S2CID 30234652.
54. ^ Sawada, Joe; Williams, Aaron (2018). "A Hamilton path for the sigma-tau problem". Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018. New Orleans, Louisiana: Society for Industrial and Applied Mathematics (SIAM). pp. 568–575. doi:10.1137/1.9781611975031.37.
55. ^ Corbett, P. F. (1992). "Rotator graphs: An efficient topology for point-to-point multiprocessor networks". IEEE Transactions on Parallel and Distributed Systems. 3 (5): 622–626. doi:10.1109/71.159045.
56. ^ a b Arndt, Jörg (2011). Matters Computational. Ideas, Algorithms, Source Code. Springer. doi:10.1007/978-3-642-14764-7. ISBN 978-3-642-14763-0.
57. ^ Alexiou, A.; Psiha, M.; Vlamos, P. (2011). "Combinatorial permutation based algorithm for representation of closed RNA secondary structures". Bioinformation. 7 (2): 91–95. doi:10.6026/97320630007091. PMC 3174042. PMID 21938211.
58. ^ "3GPP TS 36.212".
59. ^ Dolev, Shlomi; Lahiani, Limor; Haviv, Yinnon (2013). "Unique permutation hashing". Theoretical Computer Science. 475: 59–65. doi:10.1016/j.tcs.2012.12.047.

Bibliography

• Bogart, Kenneth P. (1990), Introductory Combinatorics (2nd ed.), Harcourt Brace Jovanovich, ISBN 978-0-15-541576-8
• Bóna, Miklós (2004), Combinatorics of Permutations, Chapman Hall-CRC, ISBN 978-1-58488-434-7
• Bona, Miklos (2012), Combinatorics of Permutations (2nd ed.), CRC Press, ISBN 978-1-4398-5051-0
• Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Prentice-Hall, ISBN 978-0-13-602040-0
• Cameron, Peter J. (1994), Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, ISBN 978-0-521-45761-3
• Carmichael, Robert D. (1956) [1937], Introduction to the theory of Groups of Finite Order, Dover, ISBN 978-0-486-60300-1
• Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
• Gerstein, Larry J. (1987), Discrete Mathematics and Algebraic Structures, W.H. Freeman and Co., ISBN 978-0-7167-1804-8
• Hall, Marshall Jr. (1959), The Theory of Groups, MacMillan
• Humphreys, J. F. (1996), A course in group theory, Oxford University Press, ISBN 978-0-19-853459-4
• Knuth, Donald (1973), Sorting and Searching, The Art of Computer Programming, vol. 3 This book mentions the Lehmer code (without using that name) as a variant C1,...,Cn of inversion tables in exercise 5.1.1–7 (p. 19), together with two other variants.
• Knuth, Donald (2005), Generating All Tuples and Permutations, The Art of Computer Programming, vol. 4, Addison–Wesley, ISBN 978-0-201-85393-3 Fascicle 2, first printing.
• McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225
• Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646
• Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice-Hall, ISBN 978-0-13-087868-7
• Stedman, Fabian (1677), Campanalogia, London The publisher is given as "W.S." who may have been William Smith, possibly acting as agent for the Society of College Youths, to which society the "Dedicatory" is addressed. In quotations the original long "S" has been replaced by a modern short "s".
• Webster's Seventh New Collegiate Dictionary, Springfield: G. & C. Merriam Company, 1969

Further reading

• Biggs, Norman L. (2002), Discrete Mathematics (2nd ed.), Oxford University Press, ISBN 978-0-19-850717-8
• Foata, Dominique; Schutzenberger, Marcel-Paul (1970), Théorie Géométrique des Polynômes Eulériens, Lecture Notes in Mathematics, vol. 138, Berlin, Heidelberg: Springer-Verlag, ISBN 978-3-540-04927-2. The link is to a freely available retyped (LaTeX'ed) and revised version of the text originally published by Springer-Verlag.
• Knuth, Donald (1998), Sorting and Searching, The Art of Computer Programming, vol. 3 (Second ed.), Addison–Wesley, ISBN 978-0-201-89685-5. Section 5.1: Combinatorial Properties of Permutations, pp. 11–72.
• Sedgewick, Robert (1977). "Permutation generation methods". ACM Computing Surveys. 9 (2): 137–164. doi:10.1145/356689.356692. S2CID 12139332.
• Masato, Kobayashi (2011). "Enumeration of bigrassmannian permutations below a permutation in Bruhat order". Order. 1: 131–137.

External links

• "Permutation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

Wikiversity has learning resources about Permutation notation

Wikimedia Commons has media related to Permutations.

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Standard for wireless broadband communication for mobile devices

"Long-term evolution" redirects here. For the biological concept, see Evolution and E. coli long-term evolution experiment.

In telecommunications, long-term evolution (LTE) is a standard for wireless broadband communication for mobile devices and data terminals, based on the GSM/EDGE and UMTS/HSPA standards. It improves on those standards' capacity and speed by using a different radio interface and core network improvements.[1][2] LTE is the upgrade path for carriers with both GSM/UMTS networks and CDMA2000 networks. Because LTE frequencies and bands differ from country to country, only multi-band phones can use LTE in all countries where it is supported.

The standard is developed by the 3GPP (3rd Generation Partnership Project) and is specified in its Release 8 document series, with minor enhancements described in Release 9. LTE is also called 3.95G and has been marketed as "4G LTE" and "Advanced 4G";[citation needed] but it does not meet the technical criteria of a 4G wireless service, as specified in the 3GPP Release 8 and 9 document series for LTE Advanced. The requirements were set forth by the ITU-R organisation in the IMT Advanced specification; but, because of market pressure and the significant advances that WiMAX, Evolved High Speed Packet Access, and LTE bring to the original 3G technologies, ITU later decided that LTE and the aforementioned technologies can be called 4G technologies.[3] The LTE Advanced standard formally satisfies the ITU-R requirements for being considered IMT-Advanced.[4] To differentiate LTE Advanced and WiMAX-Advanced from current 4G technologies, ITU has defined the latter as "True 4G".[5][6]

Overview

See also: List of LTE networks

LTE stands for Long-Term Evolution[7] and is a registered trademark owned by ETSI (European Telecommunications Standards Institute) for the wireless data communications technology and a development of the GSM/UMTS standards. However, other nations and companies do play an active role in the LTE project. The goal of LTE was to increase the capacity and speed of wireless data networks using new DSP (digital signal processing) techniques and modulations that were developed around the turn of the millennium. A further goal was the redesign and simplification of the network architecture to an IP-based system with significantly reduced transfer latency compared with the 3G architecture. The LTE wireless interface is incompatible with 2G and 3G networks, so that it must be operated on a separate radio spectrum. The idea of base of LTE was first proposed in 1998 , with the use of the COFDM radio access technique to replace the CDMA and studying its Terrestrial use in the L band at 1428 MHz (TE) In 2004 by Japan's NTT Docomo, with studies on the standard officially commenced in 2005.[8] In May 2007, the LTE/SAE Trial Initiative (LSTI) alliance was founded as a global collaboration between vendors and operators with the goal of verifying and promoting the new standard in order to ensure the global introduction of the technology as quickly as possible.[9][10] The LTE standard was finalized in December 2008, and the first publicly available LTE service was launched by TeliaSonera in Oslo and Stockholm on December 14, 2009, as a data connection with a USB modem. The LTE services were launched by major North American carriers as well, with the Samsung SCH-r900 being the world's first LTE Mobile phone starting on September 21, 2010,[11][12] and Samsung Galaxy Indulge being the world's first LTE smartphone starting on February 10, 2011,[13][14] both offered by MetroPCS, and the HTC ThunderBolt offered by Verizon starting on March 17 being the second LTE smartphone to be sold commercially.[15][16] In Canada, Rogers Wireless was the first to launch LTE network on July 7, 2011, offering the Sierra Wireless AirCard 313U USB mobile broadband modem, known as the "LTE Rocket stick" then followed closely by mobile devices from both HTC and Samsung.[17] Initially, CDMA operators planned to upgrade to rival standards called UMB and WiMAX, but major CDMA operators (such as Verizon, Sprint and MetroPCS in the United States, Bell and Telus in Canada, au by KDDI in Japan, SK Telecom in South Korea and China Telecom/China Unicom in China) have announced instead they intend to migrate to LTE. The next version of LTE is LTE Advanced, which was standardized in March 2011.[18] Services commenced in 2013.[19] Additional evolution known as LTE Advanced Pro have been approved in year 2015.[20]

The LTE specification provides downlink peak rates of 300 Mbit/s, uplink peak rates of 75 Mbit/s and QoS provisions permitting a transfer latency of less than 5 ms in the radio access network. LTE has the ability to manage fast-moving mobiles and supports multi-cast and broadcast streams. LTE supports scalable carrier bandwidths, from 1.4 MHz to 20 MHz and supports both frequency division duplexing (FDD) and time-division duplexing (TDD). The IP-based network architecture, called the Evolved Packet Core (EPC) designed to replace the GPRS Core Network, supports seamless handovers for both voice and data to cell towers with older network technology such as GSM, UMTS and CDMA2000.[21] The simpler architecture results in lower operating costs (for example, each E-UTRA cell will support up to four times the data and voice capacity supported by HSPA[22]).

History

3GPP standard development timeline

• In 2004, NTT Docomo of Japan proposes LTE as the international standard.[23]
• In September 2006, Siemens Networks (today Nokia Networks) showed in collaboration with Nomor Research the first live emulation of an LTE network to the media and investors. As live applications two users streaming an HDTV video in the downlink and playing an interactive game in the uplink have been demonstrated.[24]
• In February 2007, Ericsson demonstrated for the first time in the world, LTE with bit rates up to 144 Mbit/s[25]
• In September 2007, NTT Docomo demonstrated LTE data rates of 200 Mbit/s with power level below 100 mW during the test.[26]
• In November 2007, Infineon presented the world's first RF transceiver named SMARTi LTE supporting LTE functionality in a single-chip RF silicon processed in CMOS[27][28]
• In early 2008, LTE test equipment began shipping from several vendors and, at the Mobile World Congress 2008 in Barcelona, Ericsson demonstrated the world's first end-to-end mobile call enabled by LTE on a small handheld device.[29] Motorola demonstrated an LTE RAN standard compliant eNodeB and LTE chipset at the same event.
• At the February 2008 Mobile World Congress:
  • Motorola demonstrated how LTE can accelerate the delivery of personal media experience with HD video demo streaming, HD video blogging, Online gaming and VoIP over LTE running a RAN standard compliant LTE network & LTE chipset.[30]
  • Ericsson EMP (now ST-Ericsson) demonstrated the world's first end-to-end LTE call on handheld[29] Ericsson demonstrated LTE FDD and TDD mode on the same base station platform.
  • Freescale Semiconductor demonstrated streaming HD video with peak data rates of 96 Mbit/s downlink and 86 Mbit/s uplink.[31]
  • NXP Semiconductors (now a part of ST-Ericsson) demonstrated a multi-mode LTE modem as the basis for a software-defined radio system for use in cellphones.[32]
  • picoChip and Mimoon demonstrated a base station reference design. This runs on a common hardware platform (multi-mode / software-defined radio) with their WiMAX architecture.[33]
• In April 2008, Motorola demonstrated the first EV-DO to LTE hand-off – handing over a streaming video from LTE to a commercial EV-DO network and back to LTE.[34]
• In April 2008, LG Electronics and Nortel demonstrated LTE data rates of 50 Mbit/s while travelling at 110 km/h (68 mph).[35]
• In November 2008, Motorola demonstrated industry first over-the-air LTE session in 700 MHz spectrum.[36]
• Researchers at Nokia Siemens Networks and Heinrich Hertz Institut have demonstrated LTE with 100 Mbit/s Uplink transfer speeds.[37]
• At the February 2009 Mobile World Congress:
  • Infineon demonstrated a single-chip 65 nm CMOS RF transceiver providing 2G/3G/LTE functionality[38]
  • Launch of ng Connect program, a multi-industry consortium founded by Alcatel-Lucent to identify and develop wireless broadband applications.[39]
  • Motorola provided LTE drive tour on the streets of Barcelona to demonstrate LTE system performance in a real-life metropolitan RF environment[40]
• In July 2009, Nujira demonstrated efficiencies of more than 60% for an 880 MHz LTE Power Amplifier[41]
• In August 2009, Nortel and LG Electronics demonstrated the first successful handoff between CDMA and LTE networks in a standards-compliant manner[42]
• In August 2009, Alcatel-Lucent receives FCC certification for LTE base stations for the 700 MHz spectrum band.[43]
• In September 2009, Nokia Siemens Networks demonstrated world's first LTE call on standards-compliant commercial software.[44]
• In October 2009, Ericsson and Samsung demonstrated interoperability between the first ever commercial LTE device and the live network in Stockholm, Sweden.[45]
• In October 2009, Alcatel-Lucent's Bell Labs, Deutsche Telekom Innovation Laboratories, the Fraunhofer Heinrich-Hertz Institut and antenna supplier Kathrein conducted live field tests of a technology called Coordinated Multipoint Transmission (CoMP) aimed at increasing the data transmission speeds of LTE and 3G networks.[46]
• In November 2009, Alcatel-Lucent completed first live LTE call using 800 MHz spectrum band set aside as part of the European Digital Dividend (EDD).[47]
• In November 2009, Nokia Siemens Networks and LG completed first end-to-end interoperability testing of LTE.[48]
• On December 14, 2009, the first commercial LTE deployment was in the Scandinavian capitals Stockholm and Oslo by the Swedish-Finnish network operator TeliaSonera and its Norwegian brandname NetCom (Norway). TeliaSonera incorrectly branded the network "4G". The modem devices on offer were manufactured by Samsung (dongle GT-B3710), and the network infrastructure with SingleRAN technology created by Huawei (in Oslo)[49] and Ericsson (in Stockholm). TeliaSonera plans to roll out nationwide LTE across Sweden, Norway and Finland.[50] TeliaSonera used spectral bandwidth of 10 MHz (out of the maximum 20 MHz), and Single-Input and Single-Output transmission. The deployment should have provided a physical layer net bit rates of up to 50 Mbit/s downlink and 25 Mbit/s in the uplink. Introductory tests showed a TCP goodput of 42.8 Mbit/s downlink and 5.3 Mbit/s uplink in Stockholm.[51]
• In December 2009, ST-Ericsson and Ericsson first to achieve LTE and HSPA mobility with a multimode device.[52]
• In January 2010, Alcatel-Lucent and LG complete a live handoff of an end-to-end data call between LTE and CDMA networks.[53]
• In February 2010, Nokia Siemens Networks and Movistar test the LTE in Mobile World Congress 2010 in Barcelona, Spain, with both indoor and outdoor demonstrations.[54]
• In May 2010, Mobile TeleSystems (MTS) and Huawei showed an indoor LTE network at "Sviaz-Expocomm 2010" in Moscow, Russia.[55] MTS expects to start a trial LTE service in Moscow by the beginning of 2011. Earlier, MTS has received a license to build an LTE network in Uzbekistan, and intends to commence a test LTE network in Ukraine in partnership with Alcatel-Lucent.
• At the Shanghai Expo 2010 in May 2010, Motorola demonstrated a live LTE in conjunction with China Mobile. This included video streams and a drive test system using TD-LTE.[56]
• As of 12/10/2010, DirecTV has teamed up with Verizon Wireless for a test of high-speed LTE wireless technology in a few homes in Pennsylvania, designed to deliver an integrated Internet and TV bundle. Verizon Wireless said it launched LTE wireless services (for data, no voice) in 38 markets where more than 110 million Americans live on Sunday, Dec. 5.[57]
• On May 6, 2011, Sri Lanka Telecom Mobitel demonstrated 4G LTE for the first time in South Asia, achieving a data rate of 96 Mbit/s in Sri Lanka.[58]

Carrier adoption timeline

Main article: List of LTE networks

Most carriers supporting GSM or HSUPA networks can be expected to upgrade their networks to LTE at some stage. A complete list of commercial contracts can be found at:[59]

• August 2009: Telefónica selected six countries to field-test LTE in the succeeding months: Spain, the United Kingdom, Germany and the Czech Republic in Europe, and Brazil and Argentina in Latin America.[60]
• On November 24, 2009: Telecom Italia announced the first outdoor pre-commercial experimentation in the world, deployed in Torino and totally integrated into the 2G/3G network currently in service.[61]
• On December 14, 2009, the world's first publicly available LTE service was opened by TeliaSonera in the two Scandinavian capitals Stockholm and Oslo.
• On May 28, 2010, Russian operator Scartel announced the launch of an LTE network in Kazan by the end of 2010.[62]
• On October 6, 2010, Canadian provider Rogers Communications Inc announced that Ottawa, Canada's national capital, will be the site of LTE trials. Rogers said it will expand on this testing and move to a comprehensive technical trial of LTE on both low- and high-band frequencies across the Ottawa area.[63]
• On May 6, 2011, Sri Lanka Telecom Mobitel successfully demonstrated 4G LTE for the first time in South Asia, achieving a data rate of 96 Mbit/s in Sri Lanka.[64]
• On May 7, 2011, Sri Lankan Mobile Operator Dialog Axiata PLC switched on the first pilot 4G LTE Network in South Asia with vendor partner Huawei and demonstrated a download data speed up to 127 Mbit/s.[65]
• On February 9, 2012, Telus Mobility launched their LTE service initial in metropolitan areas include Vancouver, Calgary, Edmonton, Toronto and the Greater Toronto Area, Kitchener, Waterloo, Hamilton, Guelph, Belleville, Ottawa, Montreal, Québec City, Halifax and Yellowknife.[66]
• Telus Mobility has announced that it will adopt LTE as its 4G wireless standard.[67]
• Cox Communications has its first tower for wireless LTE network build-out.[68] Wireless services launched in late 2009.
• In March 2019, the Global Mobile Suppliers Association reported that there were now 717 operators with commercially launched LTE networks (broadband fixed wireless access and or mobile).[69]

The following is a list of top 10 countries/territories by 4G LTE coverage as measured by OpenSignal.com in February/March 2019.[70][71]

Rank	Country/Territory	Penetration
1	South Korea	97.5%
2	Japan	96.3%
3	Norway	95.5%
4	Hong Kong	94.1%
5	United States	93.0%
6	Netherlands	92.8%
7	Taiwan	92.8%
8	Hungary	91.4%
9	Sweden	91.1%
10	India	90.9%

For the complete list of all the countries/territories, see list of countries by 4G LTE penetration.

LTE-TDD and LTE-FDD

Long-Term Evolution Time-Division Duplex (LTE-TDD), also referred to as TDD LTE, is a 4G telecommunications technology and standard co-developed by an international coalition of companies, including China Mobile, Datang Telecom, Huawei, ZTE, Nokia Solutions and Networks, Qualcomm, Samsung, and ST-Ericsson. It is one of the two mobile data transmission technologies of the Long-Term Evolution (LTE) technology standard, the other being Long-Term Evolution Frequency-Division Duplex (LTE-FDD). While some companies refer to LTE-TDD as "TD-LTE" for familiarity with TD-SCDMA, there is no reference to that abbreviation anywhere in the 3GPP specifications.[72][73][74]

There are two major differences between LTE-TDD and LTE-FDD: how data is uploaded and downloaded, and what frequency spectra the networks are deployed in. While LTE-FDD uses paired frequencies to upload and download data,[75] LTE-TDD uses a single frequency, alternating between uploading and downloading data through time.[76][77] The ratio between uploads and downloads on a LTE-TDD network can be changed dynamically, depending on whether more data needs to be sent or received.[78] LTE-TDD and LTE-FDD also operate on different frequency bands,[79] with LTE-TDD working better at higher frequencies, and LTE-FDD working better at lower frequencies.[80] Frequencies used for LTE-TDD range from 1850 MHz to 3800 MHz, with several different bands being used.[81] The LTE-TDD spectrum is generally cheaper to access, and has less traffic.[79] Further, the bands for LTE-TDD overlap with those used for WiMAX, which can easily be upgraded to support LTE-TDD.[79]

Despite the differences in how the two types of LTE handle data transmission, LTE-TDD and LTE-FDD share 90 percent of their core technology, making it possible for the same chipsets and networks to use both versions of LTE.[79][82] A number of companies produce dual-mode chips or mobile devices, including Samsung and Qualcomm,[83][84] while operators CMHK and Hi3G Access have developed dual-mode networks in Hong Kong and Sweden, respectively.[85]

History of LTE-TDD

The creation of LTE-TDD involved a coalition of international companies that worked to develop and test the technology.[86] China Mobile was an early proponent of LTE-TDD,[79][87] along with other companies like Datang Telecom[86] and Huawei, which worked to deploy LTE-TDD networks, and later developed technology allowing LTE-TDD equipment to operate in white spaces—frequency spectra between broadcast TV stations.[73][88] Intel also participated in the development, setting up a LTE-TDD interoperability lab with Huawei in China,[89] as well as ST-Ericsson,[79] Nokia,[79] and Nokia Siemens (now Nokia Solutions and Networks),[73] which developed LTE-TDD base stations that increased capacity by 80 percent and coverage by 40 percent.[90] Qualcomm also participated, developing the world's first multi-mode chip, combining both LTE-TDD and LTE-FDD, along with HSPA and EV-DO.[84] Accelleran, a Belgian company, has also worked to build small cells for LTE-TDD networks.[91]

Trials of LTE-TDD technology began as early as 2010, with Reliance Industries and Ericsson India conducting field tests of LTE-TDD in India, achieving 80 megabit-per second download speeds and 20 megabit-per-second upload speeds.[92] By 2011, China Mobile began trials of the technology in six cities.[73]

Although initially seen as a technology utilized by only a few countries, including China and India,[93] by 2011 international interest in LTE-TDD had expanded, especially in Asia, in part due to LTE-TDD's lower cost of deployment compared to LTE-FDD.[73] By the middle of that year, 26 networks around the world were conducting trials of the technology.[74] The Global LTE-TDD Initiative (GTI) was also started in 2011, with founding partners China Mobile, Bharti Airtel, SoftBank Mobile, Vodafone, Clearwire, Aero2 and E-Plus.[94] In September 2011, Huawei announced it would partner with Polish mobile provider Aero2 to develop a combined LTE-TDD and LTE-FDD network in Poland,[95] and by April 2012, ZTE Corporation had worked to deploy trial or commercial LTE-TDD networks for 33 operators in 19 countries.[85] In late 2012, Qualcomm worked extensively to deploy a commercial LTE-TDD network in India, and partnered with Bharti Airtel and Huawei to develop the first multi-mode LTE-TDD smartphone for India.[84]

In Japan, SoftBank Mobile launched LTE-TDD services in February 2012 under the name Advanced eXtended Global Platform (AXGP), and marketed as SoftBank 4G (ja). The AXGP band was previously used for Willcom's PHS service, and after PHS was discontinued in 2010 the PHS band was re-purposed for AXGP service.[96][97]

In the U.S., Clearwire planned to implement LTE-TDD, with chip-maker Qualcomm agreeing to support Clearwire's frequencies on its multi-mode LTE chipsets.[98] With Sprint's acquisition of Clearwire in 2013,[75][99] the carrier began using these frequencies for LTE service on networks built by Samsung, Alcatel-Lucent, and Nokia.[100][101]

As of March 2013, 156 commercial 4G LTE networks existed, including 142 LTE-FDD networks and 14 LTE-TDD networks.[86] As of November 2013, the South Korean government planned to allow a fourth wireless carrier in 2014, which would provide LTE-TDD services,[77] and in December 2013, LTE-TDD licenses were granted to China's three mobile operators, allowing commercial deployment of 4G LTE services.[102]

In January 2014, Nokia Solutions and Networks indicated that it had completed a series of tests of voice over LTE (VoLTE) calls on China Mobile's TD-LTE network.[103] The next month, Nokia Solutions and Networks and Sprint announced that they had demonstrated throughput speeds of 2.6 gigabits per second using a LTE-TDD network, surpassing the previous record of 1.6 gigabits per second.[104]

Features

See also: E-UTRA

Much of the LTE standard addresses the upgrading of 3G UMTS to what will eventually be 4G mobile communications technology. A large amount of the work is aimed at simplifying the architecture of the system, as it transitions from the existing UMTS circuit + packet switching combined network, to an all-IP flat architecture system. E-UTRA is the air interface of LTE. Its main features are:

• Peak download rates up to 299.6 Mbit/s and upload rates up to 75.4 Mbit/s depending on the user equipment category (with 4×4 antennas using 20 MHz of spectrum). Five different terminal classes have been defined from a voice-centric class up to a high-end terminal that supports the peak data rates. All terminals will be able to process 20 MHz bandwidth.
• Low data transfer latencies (sub-5 ms latency for small IP packets in optimal conditions), lower latencies for handover and connection setup time than with previous radio access technologies.
• Improved support for mobility, exemplified by support for terminals moving at up to 350 km/h (220 mph) or 500 km/h (310 mph) depending on the frequency
• Orthogonal frequency-division multiple access for the downlink, Single-carrier FDMA for the uplink to conserve power.
• Support for both FDD and TDD communication systems as well as half-duplex FDD with the same radio access technology.
• Support for all frequency bands currently used by IMT systems by ITU-R.
• Increased spectrum flexibility: 1.4 MHz, 3 MHz, 5 MHz, 10 MHz, 15 MHz and 20 MHz wide cells are standardized. (W-CDMA has no option for other than 5 MHz slices, leading to some problems rolling-out in countries where 5 MHz is a commonly allocated width of spectrum so would frequently already be in use with legacy standards such as 2G GSM and cdmaOne.)
• Support for cell sizes from tens of metres radius (femto and picocells) up to 100 km (62 miles) radius macrocells. In the lower frequency bands to be used in rural areas, 5 km (3.1 miles) is the optimal cell size, 30 km (19 miles) having reasonable performance, and up to 100 km cell sizes supported with acceptable performance. In the city and urban areas, higher frequency bands (such as 2.6 GHz in EU) are used to support high-speed mobile broadband. In this case, cell sizes may be 1 km (0.62 miles) or even less.
• Support of at least 200 active data clients (connected users) in every 5 MHz cell.[105]
• Simplified architecture: The network side of E-UTRAN is composed only of eNode Bs.
• Support for inter-operation and co-existence with legacy standards (e.g., GSM/EDGE, UMTS and CDMA2000). Users can start a call or transfer of data in an area using an LTE standard, and, should coverage be unavailable, continue the operation without any action on their part using GSM/GPRS or W-CDMA-based UMTS or even 3GPP2 networks such as cdmaOne or CDMA2000.
• Uplink and downlink Carrier aggregation.
• Packet-switched radio interface.
• Support for MBSFN (multicast-broadcast single-frequency network). This feature can deliver services such as Mobile TV using the LTE infrastructure, and is a competitor for DVB-H-based TV broadcast only LTE compatible devices receives LTE signal.

Voice calls

The LTE standard supports only packet switching with its all-IP network. Voice calls in GSM, UMTS and CDMA2000 are circuit switched, so with the adoption of LTE, carriers will have to re-engineer their voice call network.[106] Four different approaches sprang up:

Main article: Voice over LTE

Main article: SRVCC

One additional approach which is not initiated by operators is the usage of over-the-top content (OTT) services, using applications like Skype and Google Talk to provide LTE voice service.[107]

Most major backers of LTE preferred and promoted VoLTE from the beginning. The lack of software support in initial LTE devices, as well as core network devices, however led to a number of carriers promoting VoLGA (Voice over LTE Generic Access) as an interim solution.[108] The idea was to use the same principles as GAN (Generic Access Network, also known as UMA or Unlicensed Mobile Access), which defines the protocols through which a mobile handset can perform voice calls over a customer's private Internet connection, usually over wireless LAN. VoLGA however never gained much support, because VoLTE (IMS) promises much more flexible services, albeit at the cost of having to upgrade the entire voice call infrastructure. VoLTE will also require Single Radio Voice Call Continuity (SRVCC) in order to be able to smoothly perform a handover to a 3G network in case of poor LTE signal quality.[109]

While the industry has seemingly standardized on VoLTE for the future, the demand for voice calls today has led LTE carriers to introduce circuit-switched fallback as a stopgap measure. When placing or receiving a voice call, LTE handsets will fall back to old 2G or 3G networks for the duration of the call.

Enhanced voice quality

To ensure compatibility, 3GPP demands at least AMR-NB codec (narrow band), but the recommended speech codec for VoLTE is Adaptive Multi-Rate Wideband, also known as HD Voice. This codec is mandated in 3GPP networks that support 16 kHz sampling.[110]

Fraunhofer IIS has proposed and demonstrated "Full-HD Voice", an implementation of the AAC-ELD (Advanced Audio Coding – Enhanced Low Delay) codec for LTE handsets.[111] Where previous cell phone voice codecs only supported frequencies up to 3.5 kHz and upcoming wideband audio services branded as HD Voice up to 7 kHz, Full-HD Voice supports the entire bandwidth range from 20 Hz to 20 kHz. For end-to-end Full-HD Voice calls to succeed, however, both the caller and recipient's handsets, as well as networks, have to support the feature.[112]

Frequency bands

See also: LTE frequency bands

The LTE standard covers a range of many different bands, each of which is designated by both a frequency and a band number:

• North America – 600, 700, 850, 1700, 1900, 2300, 2500, 2600, 3500, 5000 MHz (bands 2, 4, 5, 7, 12, 13, 14, 17, 25, 26, 29, 30, 38, 40, 41, 42, 43, 46, 48, 66, 71)
• Latin America and Caribbean – 600, 700, 800, 850, 900, 1700, 1800, 1900, 2100, 2300, 2500, 2600, 3500, 5000 MHz (bands 1, 2, 3, 4, 5, 7, 8, 12, 13, 14, 17, 20, 25, 26, 28, 29, 38, 40, 41, 42, 43, 46, 48, 66, 71)
• Europe – 450, 700, 800, 900, 1500, 1800, 2100, 2300, 2600, 3500, 3700 MHz (bands 1, 3, 7, 8, 20, 22, 28, 31, 32, 38, 40, 42, 43)[113][114]
• Asia – 450, 700, 800, 850, 900, 1500, 1800, 1900, 2100, 2300, 2500, 2600, 3500 MHz (bands 1, 3, 5, 7, 8, 11, 18, 19, 20, 21, 26, 28, 31, 38, 39, 40, 41, 42)[115]
• Africa – 700, 800, 850, 900, 1800, 2100, 2500, 2600 MHz (bands 1, 3, 5, 7, 8, 20, 28, 41)[citation needed]
• Oceania (incl. Australia[116][117] and New Zealand[118]) – 700, 800, 850, 900, 1800, 2100, 2300, 2600 MHz (bands 1, 3, 7, 8, 12, 20, 28, 40)

As a result, phones from one country may not work in other countries. Users will need a multi-band capable phone for roaming internationally.

Patents

According to the European Telecommunications Standards Institute's (ETSI) intellectual property rights (IPR) database, about 50 companies have declared, as of March 2012, holding essential patents covering the LTE standard.[119] The ETSI has made no investigation on the correctness of the declarations however,[119] so that "any analysis of essential LTE patents should take into account more than ETSI declarations."[120] Independent studies have found that about 3.3 to 5 percent of all revenues from handset manufacturers are spent on standard-essential patents. This is less than the combined published rates, due to reduced-rate licensing agreements, such as cross-licensing.[121][122][123]

See also

• 4G-LTE filter
• Comparison of wireless data standards
• E-UTRA – the radio access network used in LTE
• HSPA+ – an enhancement of the 3GPP HSPA standard
• Flat IP – flat IP architectures in mobile networks
• LTE-A
• LTE-A Pro
• LTE-U
• NarrowBand IoT (NB-IoT)
• Simulation of LTE Networks
• QoS Class Identifier (QCI) – the mechanism used in LTE networks to allocate proper Quality of Service to bearer traffic
• System architecture evolution – the re-architecturing of core networks in LTE
• VoLTE
• WiMAX – a competitor to LTE

References

1. ^ "An Introduction to LTE". 3GPP LTE Encyclopedia. Retrieved December 3, 2010.
2. ^ "Long Term Evolution (LTE): A Technical Overview" (PDF). Motorola. Retrieved July 3, 2010.
3. ^ "Newsroom • Press Release". Itu.int. Retrieved October 28, 2012.
4. ^ "ITU-R Confers IMT-Advanced (4G) Status to 3GPP LTE" (Press release). 3GPP. October 20, 2010. Retrieved May 18, 2012.
5. ^ pressinfo (October 21, 2009). "Press Release: IMT-Advanced (4G) Mobile wireless broadband on the anvil". Itu.int. Retrieved October 28, 2012.
6. ^ "Newsroom • Press Release". Itu.int. Retrieved October 28, 2012.
7. ^ ETSI Long Term Evolution Archived March 3, 2015, at the Wayback Machine page
8. ^ "Work Plan 3GPP (Release 99)". January 16, 2012. Retrieved March 1, 2012.
9. ^ "LSTI job complete". Archived from the original on January 12, 2013. Retrieved March 1, 2012.
10. ^ "LTE/SAE Trial Initiative (LSTI) Delivers Initial Results". November 7, 2007. Retrieved March 1, 2012.
11. ^ Temple, Stephen (November 18, 2014). "Vintage Mobiles: Samsung SCH-r900 – The world's first LTE Mobile (2010)". History of GMS: Birth of the mobile revolution.
12. ^ "Samsung Craft, the world's first 4G LTE phone, now available at MetroPCS". Unwired View. September 21, 2010. Archived from the original on June 10, 2013. Retrieved April 24, 2013.
13. ^ "MetroPCS debuts first 4G LTE Android phone, Samsung Galaxy Indulge". Android and Me. February 9, 2011. Retrieved March 15, 2012.
14. ^ "MetroPCS snags first LTE Android phone". Networkworld.com. Archived from the original on January 17, 2012. Retrieved March 15, 2012.
15. ^ "Verizon launches its first LTE handset". Telegeography.com. March 16, 2011. Retrieved March 15, 2012.
16. ^ "HTC ThunderBolt is officially Verizon's first LTE handset, come March 17th". Phonearena.com. Retrieved March 15, 2012.
17. ^ "Rogers lights up Canada's first LTE network today". CNW Group Ltd. July 7, 2011. Archived from the original on July 16, 2015. Retrieved October 28, 2012.
18. ^ LTE – An End-to-End Description of Network Architecture and Elements. 3GPP LTE Encyclopedia. 2009.
19. ^ "AT&T commits to LTE-Advanced deployment in 2013, Hesse and Mead unfazed". Engadget. November 8, 2011. Retrieved March 15, 2012.
20. ^ "What is LTE-Advanced Pro?". 5g.co.uk. Retrieved June 9, 2019.
21. ^ LTE – an introduction (PDF). Ericsson. 2009. Archived from the original (PDF) on August 1, 2010.
22. ^ "Long Term Evolution (LTE)" (PDF). Motorola. Retrieved April 11, 2011.
23. ^ "The Asahi Shimbun". The Asahi Shimbun. Retrieved June 9, 2019.
24. ^ "Nomor Research: World's first LTE demonstration". Archived from the original on October 5, 2011. Retrieved August 12, 2008.
25. ^ "Ericsson demonstrates live LTE at 144Mbps". Archived from the original on August 27, 2009.
26. ^ "Design". Archived from the original on September 27, 2011.
27. ^ "Infineon Ships One Billion RF-Transceivers; Introduces Next-Generation LTE Chip". Infineon Technologies. Retrieved June 9, 2019.
28. ^ "Intel® Mobile Modem Solutions". Intel. Retrieved June 9, 2019.
29. ^ a b "Ericsson to make World-first demonstration of end-to-end LTE call on handheld devices at Mobile World Congress, Barcelona". Archived from the original on September 9, 2009.
30. ^ "Motorola Media Center – Press Releases". Motorola. February 7, 2008. Retrieved March 24, 2010.
31. ^ "Freescale Semiconductor To Demo LTE In Mobile Handsets". InformationWeek.
32. ^ "Walko, John "NXP powers ahead with programmable LTE modem", EETimes, January 30, 2008".
33. ^ "Walko, John "PicoChip, MimoOn team for LTE ref design", EETimes, February 4, 2008".
34. ^ "Motorola Media Center – Press Releases". Motorola. March 26, 2008. Retrieved March 24, 2010.
35. ^ "Nortel and LG Electronics Demo LTE at CTIA and with High Vehicle Speeds:: Wireless-Watch Community". Archived from the original on June 6, 2008.
36. ^ "Motorola Media Center – – Motorola Demonstrates Industry First Over-the-Air LTE Session in 700 MHz Spectrum". Mediacenter.motorola.com. November 3, 2008. Retrieved March 24, 2010.
37. ^ "News and events". Nokia. Retrieved June 9, 2019.
38. ^ "Infineon Introduces Two New RF-Chips for LTE and 3G – SMARTi LU for Highest Data Rates with LTE and SMARTi UEmicro for Lowest Cost 3G Devices". Infineon Technologies. January 14, 2009. Retrieved March 24, 2010.
39. ^ "MWC: Alcatel-Lucent focusing on cross-industry collaboration". Telephonyonline.com. Retrieved March 24, 2010.
40. ^ "Motorola Brings LTE to Life on the Streets of Barcelona". Motorola. February 16, 2009. Retrieved March 24, 2010.
41. ^ "achieves best ever LTE transmitter efficiency". Nujira. July 16, 2009. Archived from the original on July 14, 2011. Retrieved March 24, 2010.
42. ^ "News Releases: Nortel and LG Electronics Complete World's First 3GPP Compliant Active Handover Between CDMA and LTE Networks". Nortel. August 27, 2009. Archived from the original on July 14, 2011. Retrieved March 24, 2010.
43. ^ "Alcatel-Lucent gains LTE/700 MHz certification – RCR Wireless News". Rcrwireless.com. August 24, 2009. Archived from the original on September 1, 2009. Retrieved March 24, 2010.
44. ^ "World's first LTE call on commercial software". Nokia Siemens Networks. September 17, 2009. Retrieved March 24, 2010.
45. ^ "Vivo Z1 pro Mobile – 4G/LTE – Ericsson, Samsung Make LTE Connection – Telecom News Analysis". Light Reading Group. Retrieved March 24, 2010.[permanent dead link]
46. ^ Lynnette Luna (October 17, 2009). "Alcatel-Lucent says new antenna technology boosts LTE, 3G data speeds". FierceBroadbandWireless. Archived from the original on October 20, 2009. Retrieved March 24, 2010.
47. ^ "Alcatel-Lucent completes first 800 MHz live LTE call". The Inquirer. January 11, 2010. Archived from the original on November 21, 2009. Retrieved March 24, 2010.{{cite web}}: CS1 maint: unfit URL (link)
48. ^ "and LG complete first end-to-end interoperability testing of LTE". Nokia Siemens Networks. November 24, 2009. Retrieved March 24, 2010.
49. ^ Goldstein, Phil (December 14, 2009). "TeliaSonera launches first commercial LTE network". fiercewireless.com. FierceMarkets. Retrieved October 21, 2011.
50. ^ "NetCom 4G". Archived from the original on December 20, 2012. –
51. ^ "Daily Mobile Blog". Archived from the original on April 19, 2012.
52. ^ "ST-Ericsson". ST-Ericsson. Archived from the original on January 28, 2013. Retrieved March 24, 2010.
53. ^ "Alcatel-Lucent and LG Electronics Complete a Live Handoff of an End-to-End Data Call Between LTE and CDMA networks". Your Communication News. January 8, 2010. Retrieved March 24, 2010.
54. ^ "4G Wireless Evolution – Telefonica and Nokia Siemens Demonstrate Live LTE in a Real Network Environment". Mobility Tech Zone. Technology Marketing Corp. (TMCnet). February 15, 2010. Retrieved March 24, 2010.
55. ^ "MTS and Huawei showcase LTE at Sviaz-Expocomm 2010" (in Russian). Mobile TeleSystems. May 11, 2010. Archived from the original on July 18, 2011. Retrieved May 22, 2010.
56. ^ "Front Page". The Official Motorola Blog.
57. ^ "DirecTV Tests LTE With Verizon Wireless".
58. ^ "SRI LANKA TELECOM MOBITEL RINGS IN 20 SUCCESSFUL YEARS. Well on its way to lead Sri Lanka towards an info-com and knowledge-rich society | Mobitel". www.mobitel.lk.
59. ^ "LTE Commercial Contracts". Retrieved December 10, 2010.
60. ^ "Telefónica drives the fourth-generation mobile technology by commissioning six advanced pilot trials" (PDF). Retrieved October 2, 2009.
61. ^ "Telecom accende la rete mobile di quarta generazione". Il Sole 24 ORE. Retrieved March 24, 2010.
62. ^ "Scartel to launch "$30-$40m" LTE network in Kazan". Marchmont.ru. Retrieved June 9, 2019.
63. ^ "Rogers launches first LTE technical trial in Ottawa". reuters.com. October 6, 2010.
64. ^ "Mobitel, the first in South Asia to successfully demonstrate LTE, achieving a data rate of 96 Mbps". Mobitel. Sri Lanka Telecom. May 6, 2011. Archived from the original on June 21, 2011. Retrieved June 24, 2011.
65. ^ "Dialog empowers Colombo as South Asia's first 4G LTE powered city". Daily FT. May 9, 2011. Archived from the original on May 12, 2011. Retrieved June 9, 2019.
66. ^ "About TELUS". Archived from the original on March 14, 2015. Retrieved May 31, 2016.
67. ^ "reportonbusiness.com: Wireless sales propel Telus results".
68. ^ "Cox goes with LTE-ready CDMA". Archived from the original on July 26, 2011.
69. ^ "GSA: LTE-5G Market Statistics –March 2019 Update". Retrieved April 2, 2019.
70. ^ "The State of Mobile Network Experience- Benchmarking 5G". opensignal.com. May 29, 2019. Retrieved September 6, 2019.
71. ^ Boyland, Peter (May 2019). "The State of Mobile Network Experience (PDF)" (PDF). Opensignal. Retrieved September 6, 2019.
72. ^ "Huawei rejects EU dumping, subsidy charges". China Daily (European edition). May 23, 2013. Retrieved January 9, 2014.
73. ^ a b c d e Michael Kan (January 20, 2011). "Huawei: More Trials of TD-LTE in Asia Expected". PC World. Retrieved December 9, 2013.
74. ^ a b Liau Yun Qing (June 22, 2011). "China's TD-LTE spreads across globe". ZDNet. Retrieved December 9, 2013.
75. ^ a b Dan Meyer (February 25, 2013). "MWC 2013: TD-LTE group touts successful global roaming trials". RCR Wireless News. Retrieved December 10, 2013.
76. ^ Dan Jones (October 16, 2012). "Defining 4G: What the Heck Is LTE TDD?". Light Reading. Retrieved January 9, 2014.
77. ^ a b Kim Yoo-chul (November 18, 2013). "Gov't to pick 4th mobile carrier". The Korea Times. Retrieved December 10, 2013.
78. ^ Ian Poole. "LTE-FDD, TDD, TD-LTE Duplex Schemes". Radio-electronics.com. Retrieved January 9, 2014.
79. ^ a b c d e f g Cian O'Sullivan (10 November 2010). "Nokia developing TD-LTE devices for China Mobile". GoMo News. Archived from the original on 28 March 2014. Retrieved 9 December 2013.
80. ^ Josh Taylor (December 4, 2012). "Optus to launch TD-LTE 4G network in Canberra". ZDNet. Retrieved January 9, 2014.
81. ^ Ian Poole. "LTE Frequency Bands & Spectrum Allocations". Radio-electronics.com. Retrieved January 9, 2014.
82. ^ "MWC 2013: Ericsson and China Mobile demo first dual mode HD VoLTE call based on multi-mode chipsets". Wireless – Wireless Communications For Public Services And Private Enterprises. London, UK: Noble House Media. March 4, 2013. Archived from the original on March 28, 2014. Retrieved January 9, 2014.
83. ^ Steve Costello (August 2, 2013). "GCF and GTI partner for TD-LTE device certification". Mobile World Live. Retrieved January 9, 2014.
84. ^ a b c "Qualcomm India's Dr. Avneesh Agrawal on 4G, Snapdragon and more". Digit. February 8, 2013. Retrieved December 10, 2013.
85. ^ a b "ZTE, China Mobile Hong Kong to construct LTE-TDD network". TT Magazine. July 20, 2012. Retrieved December 10, 2013.
86. ^ a b c Tan Min (May 7, 2013). "Competitors Try Curbing China Mobile's 4G Urge". Caixin Online. Caixin Media. Retrieved December 10, 2013.
87. ^ Sophie Curtis (January 4, 2012). "TD-LTE 4G standard gains momentum: ABI Research". Techworld. Retrieved December 10, 2013.
88. ^ Nick Wood (October 21, 2011). "Huawei trials white spaces TD-LTE kit". Total Telecom. Retrieved December 10, 2013.
89. ^ "Intel and Huawei set up LTE TDD lab in China". Global Telecoms Business. April 10, 2012. Retrieved December 10, 2013.[permanent dead link]
90. ^ Sharif Sakr (December 8, 2011). "Nokia Siemens promises better TD-LTE and CDMA coverage, no alarms or surprises". Engadget. Retrieved December 10, 2013.
91. ^ Kevin Fitchard (July 4, 2013). "Belgium's Accelleran aims to corner the small cell market for that other LTE". GigaOM. Retrieved December 10, 2013.
92. ^ "Ericsson, Reliance showcases first LTE-TDD ecosystem". The Indian Express. December 2, 2010. Retrieved December 9, 2013.
93. ^ "Nokia Siemens Networks TD-LTE whitepaper" (PDF). 2010. Archived from the original (PDF) on 11 June 2014. Retrieved 5 March 2014.
94. ^ "LTE TDD: network plans, commitments, trials, deployments". Telecoms.com. Retrieved December 11, 2013.
95. ^ "Huawei partners with Aero2 to launch LTE TDD/FDD commercial network". Computer News Middle East. September 21, 2011. Retrieved December 10, 2013.
96. ^ Sam Byford (February 20, 2012). "SoftBank launching 110Mbps AXGP 4G network in Japan this week". The Verge. Retrieved June 7, 2015.
97. ^ Zahid Ghadialy (February 21, 2012). "SoftBank launching 110Mbps AXGP 4G network in Japan this week". The 3G4G Blog. Retrieved June 7, 2015.
98. ^ Phil Goldstein (June 22, 2012). "Report: TD-LTE to power 25% of LTE connections by 2016". FierceWireless. Retrieved December 10, 2013.
99. ^ Rachel King (July 9, 2013). "Done deal: Sprint now owns 100 percent of Clearwire". ZDNet. Retrieved December 10, 2013.
100. ^ Kevin Fitchard (October 30, 2013). "What's igniting Spark? A look inside Sprint's super-LTE network". GigaOM. Retrieved December 10, 2013.
101. ^ Sarah Reedy (July 12, 2013). "Sprint's LTE TDD Future to Boost Current Vendors". Light Reading. Retrieved December 10, 2013.
102. ^ Richard Lai (December 4, 2013). "China finally grants 4G licenses, but still no iPhone deal for China Mobile". Engadget. Retrieved December 10, 2013.
103. ^ Ben Munson (January 31, 2014). "China Mobile, NSN Complete Live VoLTE Test on TD-LTE". Wireless Week. Archived from the original on March 5, 2016. Retrieved February 11, 2014.
104. ^ "NSN and Sprint achieves huge leap in TD-LTE network speeds". TelecomTiger. February 6, 2014. Retrieved February 11, 2014.
105. ^ "Evolution of LTE". LTE World. Retrieved October 24, 2011.
106. ^ KG, Rohde & Schwarz GmbH & Co. "Voice and SMS in LTE". www.rohde-schwarz.com. Retrieved June 9, 2019.
107. ^ Chen, Qunhui (September 2011). "Evolution and Deployment of VoLTE" (PDF). Huawei Communicate Magazine (61). Archived from the original (PDF) on November 8, 2011..
108. ^ "VoLGA whitepaper" (PDF). Retrieved June 9, 2019.
109. ^ Incorporated, Qualcomm. "Qualcomm Chipset Powers First Successful VoIP-Over-LTE Call With Single Radio Voice Call Continuity". www.prnewswire.com. Retrieved June 9, 2019.
110. ^ "LTE delivers superior voice, too" (PDF). Ericsson. Archived from the original (PDF) on September 24, 2015.
111. ^ "Fraunhofer IIS Demos Full-HD Voice Over LTE On Android Handsets". HotHardware. February 25, 2012. Retrieved June 9, 2019.
112. ^ "Firm Set to Demo HD Voice over LTE". Archived from the original on June 19, 2013.
113. ^ "EC makes official recommendation for 790–862 MHz release". October 29, 2009. Retrieved March 11, 2012.
114. ^ "Europe plans to reserve 800 MHz frequency band for LTE and WiMAX". May 16, 2010. Retrieved March 11, 2012.
115. ^ "GSMA Intelligence — Research — Hong Kong and Singapore lead LTE charge in Asia-Pacific". www.gsmaintelligence.com. Retrieved June 9, 2019.
116. ^ "Latest news on technology and innovation". Ericsson. December 5, 2016. Retrieved June 9, 2019.
117. ^ Taylor, Josh (April 14, 2011). "Optus still evaluating LTE". ZDNet. Archived from the original on March 18, 2012.
118. ^ "New Zealand 4G LTE launch". February 28, 2013.
119. ^ a b "Who Owns LTE Patents?". ipeg. March 6, 2012. Archived from the original on March 29, 2014. Retrieved March 10, 2012.
120. ^ Elizabeth Woyke (September 21, 2011). "Identifying The Tech Leaders In LTE Wireless Patents". Forbes. Retrieved March 10, 2012. Second comment by the author: "Thus, any analysis of essential LTE patents should take into account more than ETSI declarations."
121. ^ Galetovic, Alexander; Haber, Stephen; Zaretzki, Lew (September 25, 2016). "A New Dataset on Mobile Phone Patent License Royalties". Stanford University: Hoover Institution. Retrieved January 23, 2017.
122. ^ Mallinson, Keith (August 19, 2015). "On Cumulative mobile-SEP royalties" (PDF). WiseHarbor. Retrieved January 23, 2017.
123. ^ Sidak, Gregory (2016). "What Aggregate Royalty Do Manufacturers of Mobile Phones Pay to License Standard-Essential Patents" (PDF). The Criterion Journal on Innovation. Retrieved January 19, 2017.

Further reading

• Agilent Technologies, LTE and the Evolution to 4G Wireless: Design and Measurement Challenges Archived July 10, 2019, at the Wayback Machine, John Wiley & Sons, 2009 ISBN 978-0-470-68261-6
• Beaver, Paul, "What is TD-LTE?", RF&Microwave Designline, September 2011.
• E. Dahlman, H. Ekström, A. Furuskär, Y. Jading, J. Karlsson, M. Lundevall, and S. Parkvall, "The 3G Long-Term Evolution – Radio Interface Concepts and Performance Evaluation", IEEE Vehicular Technology Conference (VTC) 2006 Spring, Melbourne, Australia, May 2006
• Erik Dahlman, Stefan Parkvall, Johan Sköld, Per Beming, 3G Evolution – HSPA and LTE for Mobile Broadband, 2nd edition, Academic Press, 2008, ISBN 978-0-12-374538-5
• Erik Dahlman, Stefan Parkvall, Johan Sköld, 4G – LTE/LTE-Advanced for Mobile Broadband, Academic Press, 2011, ISBN 978-0-12-385489-6
• Sajal K. Das, John Wiley & Sons (April 2010): Mobile Handset Design, ISBN 978-0-470-82467-2.
• Sajal K. Das, John Wiley & Sons (April 2016): Mobile Terminal Receiver Design: LTE and LTE-Advanced, ISBN 978-1-1191-0730-9 .
• H. Ekström, A. Furuskär, J. Karlsson, M. Meyer, S. Parkvall, J. Torsner, and M. Wahlqvist, "Technical Solutions for the 3G Long-Term Evolution", IEEE Commun. Mag., vol. 44, no. 3, March 2006, pp. 38–45
• Mustafa Ergen, Mobile Broadband: Including WiMAX and LTE, Springer, NY, 2009
• K. Fazel and S. Kaiser, Multi-Carrier and Spread Spectrum Systems: From OFDM and MC-CDMA to LTE and WiMAX, 2nd Edition, John Wiley & Sons, 2008, ISBN 978-0-470-99821-2
• Dan Forsberg, Günther Horn, Wolf-Dietrich Moeller, Valtteri Niemi, LTE Security, Second Edition, John Wiley & Sons Ltd, Chichester 2013, ISBN 978-1-118-35558-9
• Borko Furht, Syed A. Ahson, Long Term Evolution: 3GPP LTE Radio and Cellular Technology, CRC Press, 2009, ISBN 978-1-4200-7210-5
• Chris Johnson, LTE in BULLETS, CreateSpace, 2010, ISBN 978-1-4528-3464-1
• F. Khan, LTE for 4G Mobile Broadband – Air Interface Technologies and Performance, Cambridge University Press, 2009
• Guowang Miao, Jens Zander, Ki Won Sung, and Ben Slimane, Fundamentals of Mobile Data Networks, Cambridge University Press, 2016, ISBN 1107143217
• Stefania Sesia, Issam Toufik, and Matthew Baker, LTE – The UMTS Long Term Evolution: From Theory to Practice, Second Edition including Release 10 for LTE-Advanced, John Wiley & Sons, 2011, ISBN 978-0-470-66025-6
• Gautam Siwach, Dr Amir Esmailpour, "LTE Security Potential Vulnerability and Algorithm Enhancements", IEEE Canadian Conference on Electrical and Computer Engineering (IEEE CCECE), Toronto, Canada, May 2014
• SeungJune Yi, SungDuck Chun, YoungDae lee, SungJun Park, SungHoon Jung, Radio Protocols for LTE and LTE-Advanced, Wiley, 2012, ISBN 978-1-118-18853-8
• Y. Zhou, Z. Lei and S. H. Wong, Evaluation of Mobility Performance in 3GPP Heterogeneous Networks 2014 IEEE 79th Vehicular Technology Conference (VTC Spring), Seoul, 2014, pp. 1–5.

External links

LTE at Wikipedia's sister projects

• 
  Media from Commons
• 
  Data from Wikidata

• LTE homepage from the 3GPP website
• LTE Frequently Asked Questions
• LTE Deployment Map
• A Simple Introduction to the LTE Downlink
• LTE-3GPP.info: online LTE messages decoder fully supporting Rel.14

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