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| | == Louis Vuitton København fordi det gav os noget == |
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| In [[mathematics]], the notion of '''permutation''' relates to the act of '''permuting''' (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely [[Tuple|(1,2,3)]], (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). As another example, an [[anagram]] of a word is a permutation of its letters. The study of permutations in this sense generally belongs to the field of [[combinatorics]].
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| {{anchor|n-factorial}}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''.
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| Permutations occur, in more or less prominent ways, in almost every area of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of [[sorting algorithm]]s in computer science.
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| In [[algebra]] and particularly in [[group theory]], a permutation of a [[Set (mathematics)|set]] ''S'' is defined as a [[bijection]] from ''S'' to itself (i.e., a [[map (mathematics)|map]] {{nowrap|''S'' → ''S''}} for which every element of ''S'' occurs exactly once as image value). This is related to the rearrangement of ''S'' in which each element ''s'' takes the place of the corresponding ''f''(''s''). The collection of such permutations form a [[symmetric group]]. The key to its structure is the possibility to [[composition (mathematics)|compose]] permutations: performing two given rearrangements in succession defines a third rearrangement, the composition. Permutations may [[Group action|''act'']] on composite objects by rearranging their components, or by certain replacements ([[substitution (algebra)|substitution]]s) of symbols.
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| In elementary combinatorics, the ''k''-permutations, or [[partial permutation]]s, are the sequences of ''k'' distinct elements selected from a set. When ''k'' is equal to the size of the set, these are the permutations of the set.
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| The rule to determine the number of permutations of ''n'' objects was known in Indian culture at least as early as around 1150: the [[Lilavati]] by the Indian mathematician [[Bhāskara II|Bhaskara II]] contains a passage that translates to
| | == Michael Kors Danmark initiativrig == |
| <blockquote>
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| 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.<ref>N. L. Biggs, ''The roots of combinatorics'', Historia Math. 6 (1979) 109−136</ref>
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| 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 [[Polynomial#Solving polynomial equations|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.
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| The notion of permutation is used in the following contexts.
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| ===In group theory===
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| In [[group theory]] and related areas, one considers permutations of arbitrary sets, even infinite ones. A permutation of a set ''S'' is a [[bijection]] from ''S'' to itself. This allows for permutations to be composed, which allows the definition of [[Permutation group|groups of permutations]]. If ''S'' is a finite set of ''n'' elements, then there are [[factorial|''n''!]] permutations of ''S''.
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| ===In combinatorics===
| | <li>[http://www.dmnc365.com/news/html/?100913.html http://www.dmnc365.com/news/html/?100913.html]</li> |
| [[File:Permutations with repetition.svg|thumb|Permutations of multisets]] | | |
| In [[combinatorics]], a permutation is usually understood to be a [[sequence]] containing each element from a finite set once, and only once. The concept of ''sequence'' is distinct from that of a ''set'', in that the elements of a sequence appear in some order: the sequence has a first element (unless it is empty), a second element (unless its length is less than 2), and so on. In contrast, the elements in a set have no order; {1, 2, 3} and {3, 2, 1} are different ways to denote the same set. In this sense a permutation of a finite set ''S'' of ''n'' elements is equivalent to a bijection from {1, 2, ..., ''n''} to ''S'' (in which any ''i'' is mapped to the ''i''-th element of the sequence), or to a choice of a [[total ordering]] on ''S'' (for which {{nowrap|''x'' < ''y''}} if ''x'' comes before ''y'' in the sequence). There are [[factorial|''n''!]] permutations of ''S''.
| | <li>[http://www.dotman.com/bbs/viewthread.php?tid=1390343&extra= http://www.dotman.com/bbs/viewthread.php?tid=1390343&extra=]</li> |
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| There is also a weaker meaning of the term "permutation" that is sometimes used in elementary combinatorics texts, designating those sequences in which no element occurs more than once, but without the requirement to use all elements from a given set. Indeed this use often involves considering sequences of a fixed length ''k'' of elements taken from a given set of size ''n''. These objects are also known as '''[[partial permutation]]s''' or as '''sequences without repetition''', terms that avoids confusion with the other, more common, meanings of "permutation". The number of such '''''k''-permutations of ''n''''' is denoted variously by such symbols as <sub>''n'' </sub>''P''<sub>''k''</sub>, <sup>''n''</sup>''P''<sub>''k''</sub>, ''P''<sub>''n'',''k''</sub>, or ''P''(''n'',''k''), and its value is given by the product<ref>{{cite book|author=Charalambides, Ch A.|title=Enumerative Combinatorics|publisher=CRC Press|year=2002|isbn=978-1-58488-290-9|page=42|url=http://books.google.com/books?id=PDMGA-v5G54C&pg=PA42}}</ref>
| | <li>[http://globaldialogue2013.go.tz/gdforum/activity http://globaldialogue2013.go.tz/gdforum/activity]</li> |
| | | |
| : <math>n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)</math>
| | <li>[http://www.ibrainbank.com/forum.php?mod=viewthread&tid=139696&fromuid=46315 http://www.ibrainbank.com/forum.php?mod=viewthread&tid=139696&fromuid=46315]</li> |
| | | |
| which is 0 when {{nowrap|''k'' > ''n''}}, and otherwise is equal to
| | </ul> |
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| : <math>\frac{n!}{(n-k)!}.</math>
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| The product is well defined without the assumption that ''n'' is a non-negative integer and is of importance outside combinatorics as well; it is known as the [[Pochhammer symbol]] (''n'')<sub>''k''</sub> or as the ''k''-th falling factorial power ''n''<sup><u>''k''</u> </sup> of ''n''.
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| If ''M'' is a finite [[multiset]], then a '''multiset permutation''' is a sequence of elements of ''M'' in which each element appears exactly as often as is its multiplicity in ''M''. If the multiplicities of the elements of ''M'' (taken in some order) are <math>m_1</math>, <math>m_2</math>, ..., <math>m_l</math> and their sum (i.e., the size of ''M'') is ''n'', then the number of multiset permutations of ''M'' is given by the [[Multinomial_coefficient#Multinomial_coefficients|multinomial coefficient]]
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| :<math>{n \choose m_1,m_2,\ldots,m_l} = \frac{n!}{m_1!\,m_2!\, \cdots\,m_l!}.</math>
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| ==Permutations in group theory==
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| {{Main|Symmetric group}}
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| In group theory, the term ''permutation'' of a set means a bijective map, or [[bijection]], from that set onto itself. The set of all permutations of any given set ''S'' forms a [[Group (mathematics)|group]], with composition of maps as product and the identity as neutral element. This is the '''[[symmetric group]]''' of ''S''. Up to isomorphism, this symmetric group only depends on the [[cardinality]] of the set, so the nature of elements of ''S'' is irrelevant for the structure of the group. Symmetric groups have been studied most in the case of a finite sets, in which case one can assume without loss of generality that ''S''={1,2,...,''n''} for some natural number ''n'', which defines the symmetric group of degree ''n'', written {{math|'''S'''<sub>''n''</sub>}}.
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| Any subgroup of a symmetric group is called a '''[[permutation group]]'''. In fact by [[Cayley's theorem]] any group is isomorphic to some permutation group, and every finite group to a subgroup of some finite symmetric group. However, permutation groups have more structure than abstract groups, allowing for instance to define the [[cycle type]] of an element of a permutation group; different realizations of a group as a permutation group need not be equivalent for this additional structure. For instance {{math|'''S'''<sub>3</sub>}} is naturally a permutation group, in which any transposition has cycle type (2,1), but the proof of Cayley's theorem realizes {{math|'''S'''<sub>3</sub>}} as a subgroup of {{math|'''S'''<sub>6</sub>}} (namely the permutations of the 6 elements of {{math|'''S'''<sub>3</sub>}} itself), in which permutation group transpositions get cycle type (2,2,2). So in spite of Cayley's theorem, the study of permutation groups differs from the study of abstract groups.
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| ===Notation=== <!-- Redirect [[one-line notation]] links here -->
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| There are three main notations for permutations of a finite set ''S''. In [[Augustin-Louis Cauchy|Cauchy]]'s ''two-line notation'',<ref>{{Cite book|title=The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory|first=Hans|last=Wussing|publisher=Courier Dover Publications|year=2007|isbn=9780486458687|page=94|url=http://books.google.com/books?id=Xp3JymnfAq4C&pg=PA94|quote=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|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref> one lists the elements of ''S'' in the first row, and for each one its image under the permutation below it in the second row. For instance, a particular permutation of the set {1,2,3,4,5} can be written as:
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| : <math>\sigma=\begin{pmatrix}
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| 1 & 2 & 3 & 4 & 5 \\
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| 2 & 5 & 4 & 3 & 1\end{pmatrix};</math>
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| this means that ''σ'' satisfies ''σ''(1)=2, ''σ''(2)=5, ''σ''(3)=4, ''σ''(4)=3, and ''σ''(5)=1.
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| In ''one-line notation'', one gives only the second row of this array, so the one-line notation for the permutation above is 25431. (It is typical to use commas to separate these entries only if some have two or more digits.)
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| ''[[Cycle notation]]'', the third method of notation, focuses on the effect of successively applying parts of the permutation. It expresses the permutation as a product of [[cycle (mathematics)|cycle]]s corresponding to the [[orbit (group theory)|orbit]]s (with at least two elements) of the permutation; since distinct orbits are [[disjoint sets|disjoint]], this is loosely referred to as "the decomposition into disjoint cycles" of the permutation. It works as follows: starting from some element ''x'' of ''S'' with {{nowrap|''σ''(''x'') ≠ ''x''}}, one writes the sequence (''x'' ''σ''(''x'') ''σ''(''σ''(''x'')) ...) of successive images under ''σ'', until the image would be ''x'', at which point one instead closes the parenthesis. The set of values written down forms the orbit (under ''σ'') of ''x'', and the parenthesized expression gives the corresponding cycle of ''σ''. One then continues choosing an element ''y'' of ''S'' that is not in the orbit already written down, and such that {{nowrap|''σ''(''y'') ≠ ''y''}}, and writes down the corresponding cycle, and so on until all elements of ''S'' either belong to a cycle written down or are [[fixed point (mathematics)|fixed point]]s of ''σ''. Since for every new cycle the starting point can be chosen in different ways, there are in general many different cycle notations for the same permutation; for the example above one has for instance
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| :<math>\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\
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| 2 & 5 & 4 & 3 & 1\end{pmatrix}=\begin{pmatrix}1 & 2 & 5 \end{pmatrix} \begin{pmatrix}3 & 4 \end{pmatrix} = \begin{pmatrix}3 & 4 \end{pmatrix} \begin{pmatrix}1 & 2 & 5 \end{pmatrix} = \begin{pmatrix}3 & 4 \end{pmatrix} \begin{pmatrix}5 & 1 & 2 \end{pmatrix}.</math>
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| Each cycle (''x''<sub>1</sub> ''x''<sub>2</sub> ... ''x''<sub>''l''</sub>) of ''σ'' denotes a permutation in its own right, namely the one that takes the same values as ''σ'' on this orbit (so it maps ''x''<sub>''i''</sub> to ''x''<sub>''i''+1</sub> for {{nowrap|''i'' < ''l''}}, and ''x''<sub>''l''</sub> to ''x''<sub>1</sub>), while mapping all other elements of ''S'' to themselves. The size ''l'' of the orbit is called the length of the cycle. Distinct orbits of ''σ'' are by definition disjoint, so the corresponding cycles [[commutativity|commute]], and ''σ'' is the product of its cycles (taken in any order). Therefore the concatenation of cycles in the cycle notation can be interpreted as denoting composition of permutations, whence the name "decomposition" of the permutation. This decomposition is essentially unique: apart from the reordering the cycles in the product, there are no other ways to write ''σ'' as a product of cycles (possibly unrelated to the cycles of ''σ'') that have disjoint orbits. The cycle notation is less unique, since each individual cycle can be written in different ways, as in the example above where (5 1 2) denotes the same cycle as (1 2 5) or (2 5 1) (though note that (5 2 1) denotes a different cycle).
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| An orbit of size 1 (a fixed point ''x'' in ''S'') has no corresponding cycle, since that permutation would fix ''x'' as well as every other element of ''S'', in other words it would be the identity, independently of ''x''. It is possible to include (''x'') in the cycle notation for ''σ'' to stress that ''σ'' fixes ''x'' (and this is even standard in [[combinatorics]], as described in [[cycles and fixed points]]), but this does not correspond to a factor in the (group theoretic) decomposition of ''σ''. If the notion of "cycle" were taken to include the identity permutation, then this would spoil the uniqueness (up to order) of the decomposition of a permutation into disjoint cycles. The decomposition into disjoint cycles of the identity permutation is an [[empty product]]; its cycle notation would be empty, so some other notation like ''e'' is usually used instead.
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| Cycles of length two are called [[transposition (mathematics)|transpositions]]; such permutations merely exchange the place of two elements.
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| ===Group structure===
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| {{Main|Symmetric group}}
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| ====Product and inverse====
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| The product of two permutations is defined as their [[function composition|composition]] as functions, in other words ''σ·π'' is the function that maps any element ''x'' of the set to ''σ''(''π''(''x'')). Note that the rightmost permutation is applied to the argument first,
| |
| <ref>
| |
| {{cite book | last1=Biggs | first1=Norman L. | last2=White | first2=A. T.
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| |year=1979
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| |publisher=Cambridge University Press
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| |title=Permutation groups and combinatorial structures
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| |isbn=0-521-22287-7
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| }}
| |
| </ref>
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| because of the way function application is written. Some authors prefer the leftmost factor acting first,
| |
| <ref>
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| {{cite book | last1=Dixon | first1=John D. | last2=Mortimer | first2=Brian
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| |title =Permutation Groups
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| |isbn=0-387-94599-7
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| |publisher=Springer
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| |year=1996
| |
| }}
| |
| </ref>
| |
| <ref>
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| {{cite book | last1=Cameron | first1= Peter J.
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| |year=1999
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| |publisher=Cambridge University Press
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| |title=Permutation groups
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| |isbn=0-521-65302-9
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| }}
| |
| </ref>
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| <ref>
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| {{cite journal | first1=M. | last1=Jerrum
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| |journal = J. Algor.
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| |volume=7
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| |number=1
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| |title=A compact representation of permutation groups
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| |pages=60–78
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| |year=1986
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| }}
| |
| </ref>
| |
| but to that end permutations must be written to the ''right'' of their argument, for instance as an exponent, where ''σ'' acting on ''x'' is written ''x''<sup>''σ''</sup>; then the product is defined by ''x''<sup>''σ·π''</sup> = (''x''<sup>''σ''</sup>)<sup>''π''</sup>. However this gives a ''different'' rule for multiplying permutations; this article uses the definition where the rightmost permutation is applied first.
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| Since the composition of two bijections always gives another bijection, the product of two permutations is again a permutation. Since [[function composition]] is [[associative]], so is the product operation on permutations: (''σ·π'')·''ρ'' = ''σ''·(''π·ρ''). Therefore, products of more than two permutations are usually written without adding parentheses to express grouping; they are also usually written without a dot or other sign to indicate multiplication.
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| The identity permutation, which maps every element of the set to itself, is the neutral element for this product. In two-line notation, the identity is
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| :<math>\begin{pmatrix}1 & 2 & 3 & \cdots & n \\ 1 & 2 & 3 & \cdots & n\end{pmatrix}.</math>
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| Since [[bijections]] have [[Inverse function|inverses]], so do permutations, and the inverse ''σ''<sup>−1</sup> of ''σ'' is again a permutation. Explicitly, whenever ''σ''(''x'')=''y'' one also has ''σ''<sup>−1</sup>(''y'')=''x''. In two-line notation the inverse can be obtained by interchanging the two lines (and sorting the columns if one wishes the first line to be in a given order). For instance
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| :<math>\begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix}^{-1}
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| =\begin{pmatrix}2 & 5 & 4 & 3 & 1\\ 1 & 2 & 3 & 4 & 5 \end{pmatrix}
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| =\begin{pmatrix}1 & 2 & 3 & 4 & 5 \\ 5 & 1 & 4 & 3 & 2\end{pmatrix}.</math>
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| In cycle notation one can reverse the order of the elements in each cycle to obtain a cycle notation for its inverse.
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| Having an associative product, a neutral element, and inverses for all its elements, makes the set of all permutations of ''S'' into a [[group (mathematics)|group]], called the symmetric group of ''S''.
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| ====Properties====
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| Every permutation of a finite set can be expressed as the product of transpositions.
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| Moreover, although many such expressions for a given permutation may exist, there can never be among them both expressions with an even number and expressions with an odd number of transpositions. All permutations are then classified as [[Even and odd permutations|even or odd]], according to the parity of the transpositions in any such expression.
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| [[File:Symmetric group 3; Cayley table; matrices.svg|thumb|Composition of permutations corresponds to multiplication of permutation matrices.]]
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| Multiplying permutations written in cycle notation follows no easily described pattern, and the cycles of the product can be entirely different from those of the permutations being composed. However the cycle structure is preserved in the special case of [[conjugacy class|conjugating]] a permutation ''σ'' by another permutation ''π'', which means forming the product ''π·σ·π''<sup>−1</sup>. Here the cycle notation of the result can be obtained by taking the cycle notation for ''σ'' and applying ''π'' to all the entries in it.<ref>{{Google books quote|id=2jBqvVb0Q-AC|page=84|text=conjugate permutations have the same cycle type|Humphreys (1996), p. 84}}</ref>
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| ====Matrix representation====
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| One can represent a permutation of {1, 2, ..., ''n''} as an ''n''×''n'' [[matrix (mathematics)|matrix]]. There are two natural ways to do so, but only one for which multiplications of matrices corresponds to multiplication of permutations in the same order: this is the one that associates to ''σ'' the matrix ''M'' whose entry ''M''<sub>''i'',''j''</sub> is 1 if ''i'' = ''σ''(''j''), and 0 otherwise. The resulting matrix has exactly one entry 1 in each column and in each row, and is called a ''[[permutation matrix]]''. <br>
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| [http://upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Symmetric_group_4%3B_permutation_list_with_matrices.svg/1000px-Symmetric_group_4%3B_permutation_list_with_matrices.svg.png Here] <small>([[:File:Symmetric group 4; permutation list with matrices.svg|file]])</small> is a list of these matrices for permutations of 4 elements. The [[Cayley table]] on the right shows these matrices for permutations of 3 elements.
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| ====Permutation of components of a sequence====
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| As with any group, one can consider actions of a symmetric group on a set, and there are many ways in which such an action can be defined. For the symmetric group of {1, 2, ..., ''n''} there is one particularly natural action, namely the action by permutation on the set ''X''<sup>''n''</sup> of sequences of ''n'' symbols taken from some set ''X''. Like for the matrix representation, there are two natural ways in which the result of permuting a sequence (''x''<sub>1</sub>,''x''<sub>2</sub>,...,''x''<sub>''n''</sub>) by ''σ'' can be defined, but only one is compatible with the multiplication of permutations (so as to give a left action of the symmetric group on ''X''<sup>''n''</sup>); with the multiplication rule used in this article this is the one given by
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| :<math>\sigma\cdot(x_1,\ldots,x_n) = (x_{\sigma^{-1}(1)},\ldots,x_{\sigma^{-1}(n)}).</math>
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| This means that each component ''x''<sub>''i''</sub> ends up at position ''σ''(''i'') in the sequence permuted by ''σ''.
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| ==Permutations in combinatorics==
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| In combinatorics a permutation of a set ''S'' with ''n'' elements is a listing of the elements of ''S'' in some order (each element occurring exactly once). This can be defined formally as a bijection from the set { 1, 2, ..., ''n'' } to ''S''. Note that if ''S'' equals { 1, 2, ..., ''n'' }, then this definition coincides with the definition in group theory. More generally one could use instead of { 1, 2, ..., ''n'' } any set equipped with a total ordering of its elements.
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| One combinatorial property that is related to the group theoretic interpretation of permutations, and can be defined without using a total ordering of ''S'', is the [[cycles and fixed points|cycle structure]] of a permutation ''σ''. It is the [[partition (number theory)|partition]] of ''n'' describing the lengths of the cycles of ''σ''. Here there is a part "1" in the partition for every fixed point of ''σ''. A permutation that has no fixed point is called a [[derangement]].
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| Other combinatorial properties however are directly related to the ordering of ''S'', and to the way the permutation relates to it. Here are a number of such properties.
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| ===Ascents, descents and runs===
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| An ''ascent'' of a permutation ''σ'' of ''n'' is any position ''i'' < ''n'' where the following value is bigger than the current one. That is, if ''σ'' = ''σ''<sub>1</sub>''σ''<sub>2</sub>...''σ''<sub>''n''</sub>, then ''i'' is an ascent if ''σ''<sub>''i''</sub> < ''σ''<sub>''i''+1</sub>.
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| For example, the permutation 3452167 has ascents (at positions) 1,2,5,6.
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| Similarly, a ''descent'' is a position ''i'' < ''n'' with ''σ''<sub>''i''</sub> > ''σ''<sub>''i''+1</sub>, so every ''i'' with <math>1\leq i<n</math> either is an ascent or is a descent of ''σ''.
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| The number of permutations of ''n'' with ''k'' ascents is the [[Eulerian number]] <math>\textstyle\left\langle{n\atop k}\right\rangle</math>; this is also the number of permutations of ''n'' with ''k'' descents.<ref>Combinatorics of Permutations, ISBN 1-58488-434-7, M. Bóna, 2004, p. 3</ref>
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| 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.
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| For example, the permutation 2453167 has the ascending runs 245, 3, and 167, while it has an increasing subsequence 2367.
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| If a permutation has ''k'' − 1 descents, then it must be the union of ''k'' ascending runs. Hence, the number of permutations of ''n'' with ''k'' ascending runs is the same as the number <math>\textstyle\left\langle{n\atop k-1}\right\rangle</math> of permutations with ''k'' − 1 descents.<ref>Combinatorics of Permutations, ISBN 1-58488-434-7, M. Bóna, 2004, p. 4f</ref>
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| ===Inversions===
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| {{main|Inversion (discrete mathematics)}}
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| An ''[[inversion (discrete mathematics)|inversion]]'' of a permutation ''σ'' is a pair (''i'',''j'') of positions where the entries of a permutation are in the opposite order: <math>i<j</math> and <math>\sigma_i>\sigma_j</math>.<ref>Combinatorics of Permutations, ISBN 1-58488-434-7, M. Bóna, 2004, p. 43</ref> So a descent is just an inversion at two adjacent positions. For example, the permutation ''σ'' = 23154 has three inversions: (1,3), (2,3), (4,5), for the pairs of entries (2,1), (3,1), (5,4).
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| Sometimes an inversion is defined as the pair of values (''σ''<sub>''i''</sub>,''σ''<sub>''j''</sub>) itself whose order is reversed; this makes no difference for the ''number'' of inversions, and this pair (reversed) is also an inversion in the above sense for the inverse permutation ''σ''<sup>−1</sup>. 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''<sup>−1</sup>. To bring a permutation with ''k'' inversions into order (i.e., transform it into the identity permutation), by successively applying (right-multiplication by) [[adjacent transposition]]s, 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 {{nowrap|''i'' + 1}} where ''i'' is a descent of the permutation as modified so far (so that the transposition will remove this particular descent, although it might create other descents). This is so because applying such a transposition reduces the number of inversions by 1; also note that as long as this number is not zero, the permutation is not the identity, so it has at least one descent. [[Bubble sort]] and [[insertion sort]] can be interpreted as particular instances of this procedure to put a sequence into order. Incidentally this procedure proves that any permutation ''σ'' can be written as a product of adjacent transpositions; for this one may simply reverse any sequence of such transpositions that transforms ''σ'' into the identity. In fact, by enumerating all sequences of adjacent transpositions that would transform ''σ'' into the identity, one obtains (after reversal) a ''complete'' list of all expressions of minimal length writing ''σ'' as a product of adjacent transpositions.
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| The number of permutations of ''n'' with ''k'' inversions is expressed by a Mahonian number,<ref>Combinatorics of Permutations, ISBN 1-58488-434-7, M. Bóna, 2004, p. 43ff</ref> it is the coefficient of ''X''<sup>''k''</sup> in the expansion of the product
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| :<math>\prod_{m=1}^n\sum_{i=0}^{m-1}X^i=1(1+X)(1+X+X^2)\cdots(1+X+X^2+\cdots+X^{n-1}),</math>
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| which is also known (with ''q'' substituted for ''X'') as the [[q-factorial]] <nowiki>[</nowiki>''n''<nowiki>]</nowiki><sub>''q''</sub>! . The expansion of the product appears in [[Necklace (combinatorics)]].
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| ===Counting sequences without repetition===
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| In this section, a ''k''-permutation of a set ''S'' is an ordered sequence of ''k'' distinct elements of ''S''. For example, given the set of letters {<tt>C</tt>, <tt>E</tt>, <tt>G</tt>, <tt>I</tt>, <tt>N</tt>, <tt>R</tt>}, the sequence <tt>ICE</tt> is a 3-permutation, <tt>RING</tt> and <tt>RICE</tt> are 4-permutations, <tt>NICER</tt> and <tt>REIGN</tt> are 5-permutations, and <tt>CRINGE</tt> is a 6-permutation; since the latter uses all letters, it is a permutation of the given set in the ordinary combinatorial sense. <tt>ENGINE</tt> on the other hand is not a permutation, because of the repetitions: it uses the elements <tt>E</tt> and <tt>N</tt> twice.
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| Let ''n'' be the size of ''S'', the number of elements available for selection. In constructing a ''k''-permutation, there are ''n'' possible choices for the first element of the sequence, and this is then number of 1-permutations. Once it has been chosen, there are {{nowrap|''n'' − 1}} elements of ''S'' left to choose from, so a second element can be chosen in {{nowrap|''n'' − 1}} ways, giving a total ''n'' × (''n'' − 1) possible 2-permutations. For each successive element of the sequence, the number of possibilities decreases by 1 which leads to the number of
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| :''n'' × (''n'' − 1) × (''n'' − 2) ... × (''n'' − ''k'' + 1) possible ''k''-permutations.
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| This gives in particular the number of ''n''-permutations (which contain all elements of ''S'' once, and are therefore simply permutations of ''S''):
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| :''n'' × (''n'' − 1) × (''n'' − 2) × ... × 2 × 1,
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| a number that occurs so frequently in mathematics that it is given a compact notation "''n''!", and is called "''n'' [[factorial]]". These ''n''-permutations are the longest sequences without repetition of elements of ''S'', which is reflected by the fact that the above formula for the number of ''k''-permutations gives zero whenever ''k'' > ''n''.
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| The number of ''k''-permutations of a set of ''n'' elements is sometimes denoted by ''P''(''n'',''k'') or a similar notation (usually accompanied by a notation for the number of ''k''-[[combination]]s of a set of ''n'' elements in which the "''P''" is replaced by "''C''"). That notation is rarely used in other contexts than that of counting ''k''-permutations, but the expression for the number does arise in many other situations. Being a product of ''k'' factors starting at ''n'' and decreasing by unit steps, it is called the ''k''-th falling factorial power of ''n'':
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| :<math>n^{\underline k}=n\times(n-1)\times(n-2)\times\cdots\times(n-k+1),</math>
| |
| though many other names and notations are in use, as detailed at [[Pochhammer symbol]]. When ''k'' ≤ ''n'' the factorial power can be completed by additional factors: ''n''<sup><u>''k''</u></sup> × (''n'' − ''k'')! = ''n''!, which allows writing
| |
| :<math>n^{\underline k}=\frac{n!}{(n-k)!}.</math>
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| The right hand side is often given as expression for the number of ''k''-permutations, but its main merit is using the compact factorial notation. Expressing a product of ''k'' factors as a quotient of potentially much larger products, where all factors in the denominator are also explicitly present in the numerator, is not particularly efficient; as a method of computation there is the additional danger of overflow or rounding errors. It should also be noted that the expression is undefined when ''k'' > ''n'', whereas in those cases the number ''n''<sup><u>''k''</u></sup> of ''k''-permutations is just 0.
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| | |
| ==Permutations in computing==
| |
| | |
| ===Numbering permutations===
| |
| | |
| One way to represent permutations of ''n'' is by an integer ''N'' with 0 ≤ ''N'' < ''n''!, provided convenient methods are given to convert between the number and the usual representation of a permutation as a sequence. This gives the most compact representation of arbitrary permutations, and in computing is particularly attractive when ''n'' is small enough that ''N'' can be held in a machine word; for 32-bit words this means ''n'' ≤ 12, and for 64-bit words this means ''n'' ≤ 20. The conversion can be done via the intermediate form of a sequence of numbers ''d''<sub>''n''</sub>, ''d''<sub>''n''−1</sub>, ..., ''d''<sub>2</sub>, ''d''<sub>1</sub>, where ''d''<sub>''i''</sub> is a non-negative integer less than ''i'' (one may omit ''d''<sub>1</sub>, as it is always 0, but its presence makes the subsequent conversion to a permutation easier to describe). The first step then is simply expression of ''N'' in the '''[[factorial number system]]''', which is just a particular [[mixed radix]] representation, where for numbers up to ''n''! the bases for successive digits are ''n'', {{nowrap|''n'' − 1}}, ..., 2, 1. The second step interprets this sequence as a [[Lehmer code]] or (almost equivalently) as an inversion table.
| |
| | |
| {| class="wikitable" style="float:right; text-align:center; margin: 1em"
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| |+ Rothe diagram for <math>\sigma=(6,3,8,1,4,9,7,2,5)</math>
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| |-
| |
| ! <sub>''i''</sub> ＼ <sup>''σ''<sub>''i''</sub></sup>
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| ! style="width:20pt;"| 1
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| ! style="width:20pt;"| 2
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| ! style="width:20pt;"| 3
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| ! style="width:20pt;"| 4
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| ! style="width:20pt;"| 5
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| ! style="width:20pt;"| 6
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| ! style="width:20pt;"| 7
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| ! style="width:20pt;"| 8
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| ! style="width:20pt;"| 9
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| ! Lehmer code
| |
| |-
| |
| ! 1
| |
| | × || × || × || × || × || • || || || || ''d''<sub>9</sub> = 5
| |
| |-
| |
| ! 2
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| | × || × || • || || || || || || || ''d''<sub>8</sub> = 2
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| |-
| |
| ! 3
| |
| | × || × || || × || × || || × || • || || ''d''<sub>7</sub> = 5
| |
| |-
| |
| ! 4
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| | • || || || || || || || || || ''d''<sub>6</sub> = 0
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| |-
| |
| ! 5
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| | || × || || • || || || || || || ''d''<sub>5</sub> = 1
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| |-
| |
| ! 6
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| | || × || || || × || || × || || • || ''d''<sub>4</sub> = 3
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| |-
| |
| ! 7
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| | || × || || || × || || • || || || ''d''<sub>3</sub> = 2
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| |-
| |
| ! 8
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| | || • || || || || || || || || ''d''<sub>2</sub> = 0
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| |-
| |
| ! 9
| |
| | || || || || • || || || || || ''d''<sub>1</sub> = 0
| |
| |-
| |
| ! inversion table
| |
| | 3 || 6 || 1 || 2 || 4 || 0 || 2 || 0 || 0 ||
| |
| |}
| |
| In the '''Lehmer code''' for a permutation ''σ'', the number ''d''<sub>''n''</sub> represents the choice made for the first term ''σ''<sub>1</sub>, the number ''d''<sub>''n''−1</sub> represents the choice made for the second term
| |
| ''σ''<sub>2</sub> among the remaining {{nowrap|''n'' − 1}} elements of the set, and so forth. More precisely, each ''d''<sub>''n''+1−''i''</sub> gives the number of ''remaining'' elements strictly less than the term ''σ''<sub>''i''</sub>. Since those remaining elements are bound to turn up as some later term ''σ''<sub>''j''</sub>, the digit ''d''<sub>''n''+1−''i''</sub> counts the ''inversions'' (''i'',''j'') involving ''i'' as smaller index (the number of values ''j'' for which ''i'' < ''j'' and ''σ''<sub>''i''</sub> > ''σ''<sub>''j''</sub>). The '''inversion table''' for ''σ'' is quite similar, but here ''d''<sub>''n''+1−''k''</sub> counts the number of inversions (''i'',''j'') where ''k'' = ''σ''<sub>''j''</sub> occurs as the smaller of the two values appearing in inverted order.<ref name="Knuth">D. E. Knuth, ''The Art of Computer Programming'', Vol 3, Sorting and Searching, Addison–Wesley (1973), p. 12. This book mentions the Lehmer code (without using that name) as a variant ''C''<sub>1</sub>,...,''C''<sub>''n''</sub> of inversion tables in exercise 5.1.1−7 (p. 19), together with two other variants.</ref> Both encodings can be visualized by an ''n'' by ''n'' '''Rothe diagram'''{{#tag:ref|[[Heinrich August Rothe|H. A. Rothe]], ''Sammlung combinatorisch-analytischer Abhandlungen'' '''2''' (Leipzig, 1800), 263−305. Cited in,<ref name="Knuth"/> p. 14.}} (named after [[Heinrich August Rothe]]) in which dots at (''i'',''σ''<sub>''i''</sub>) mark the entries of the permutation, and a cross at (''i'',''σ''<sub>''j''</sub>) marks the inversion (''i'',''j''); by the definition of inversions a cross appears in any square that comes both before the dot (''j'',''σ''<sub>''j''</sub>) in its column, and before the dot (''i'',''σ''<sub>''i''</sub>) in its row. The Lehmer code lists the numbers of crosses in successive rows, while the inversion table lists the numbers of crosses in successive columns; it is just the Lehmer code for the inverse permutation, and vice versa.
| |
| | |
| To effectively convert a Lehmer code ''d''<sub>''n''</sub>, ''d''<sub>''n''−1</sub>, ..., ''d''<sub>2</sub>, ''d''<sub>1</sub> into a permutation of an ordered set ''S'', one can start with a list of the elements of ''S'' in increasing order, and for ''i'' increasing from 1 to ''n'' set ''σ''<sub>''i''</sub> to the element in the list that is preceded by ''d''<sub>''n''+1−''i''</sub> other ones, and remove that element from the list. To convert an inversion table ''d''<sub>''n''</sub>, ''d''<sub>''n''−1</sub>, ..., ''d''<sub>2</sub>, ''d''<sub>1</sub> into the corresponding permutation, one can traverse the numbers from ''d''<sub>1</sub> to ''d''<sub>''n''</sub> 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 (permutation)|signature]] of the permutation. Moreover the positions of the zeroes in the inversion table give the values of left-to-right maxima of the permutation (in the example 6, 8, 9) while the positions of the zeroes in the Lehmer code are the positions of the right-to-left minima (in the example positions the 4, 8, 9 of the values 1, 2, 5); this allows computing the distribution of such extrema among all permutations. A permutation with Lehmer code ''d''<sub>''n''</sub>, ''d''<sub>''n''−1</sub>, ..., ''d''<sub>2</sub>, ''d''<sub>1</sub> has an ascent {{nowrap|''n'' − ''i''}} if and only if {{nowrap|''d''<sub>''i''</sub> ≥ ''d''<sub>''i+1''</sub>}}.
| |
| | |
| ===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 data structure|array]] or a [[linked list]], both require (for different reasons) about ''n''<sup>2</sup>/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 [[big O notation|''O''(''n'' log ''n'')]] time.
| |
| | |
| ====Random generation of permutations====
| |
| | |
| {{Main|Fisher–Yates shuffle}}
| |
| | |
| For generating random permutations of a given sequence of ''n'' values, it makes no difference whether one means apply a randomly selected permutation of ''n'' to the sequence, or choose a random element from the set of distinct (multiset) permutations of the sequence. This is because, even though in case of repeated values there can be many distinct permutations of ''n'' that result in the same permuted sequence, the number of such permutations is the same for each possible result. Unlike for systematic generation, which becomes unfeasible for large ''n'' due to the growth of the number ''n''!, there is no reason to assume that ''n'' will be small for random generation.
| |
| | |
| The basic idea to generate a random permutation is to generate at random one of the ''n''! sequences of integers ''d''<sub>1</sub>,''d''<sub>2</sub>,...,''d''<sub>''n''</sub> satisfying {{nowrap|0 ≤ ''d''<sub>''i''</sub> < ''i''}} (since ''d''<sub>1</sub> is always zero it may be omitted) and to convert it to a permutation through a [[bijective]] correspondence. For the latter correspondence one could interpret the (reverse) sequence as a Lehmer code, and this gives a generation method first published in 1938 by [[Ronald A. Fisher]] and [[Frank Yates]].<ref>{{cite book
| |
| | author = Fisher, R.A.; Yates, F.
| |
| | title = Statistical tables for biological, agricultural and medical research
| |
| | origyear = 1938
| |
| | edition = 3rd
| |
| | year = 1948
| |
| | pages = 26–27
| |
| | publisher = Oliver & Boyd
| |
| | location = London
| |
| | oclc = 14222135
| |
| }}</ref>
| |
| While at the time computer implementation was not an issue, this method suffers from the difficulty sketched above to convert from Lehmer code to permutation efficiently. This can be remedied by using a different bijective correspondence: after using ''d''<sub>''i''</sub> 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 [[swap (computer science)|swap]]s 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 (mathematics)|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]]:
| |
| | |
| <blockquote>
| |
| :'''for''' ''i'' '''from''' ''n'' '''downto''' 2
| |
| :'''do''' ''d<sub>i</sub>'' ← random element of { 0, ..., ''i'' − 1 }
| |
| :: '''swap''' ''a''[''d<sub>i</sub>''] and ''a''[''i'' − 1]
| |
| </blockquote>
| |
| | |
| This can be combined with the initialization of the array ''a''[''i''] = ''i'' as follows:
| |
| | |
| <blockquote>
| |
| :'''for''' ''i'' '''from''' 0 '''to''' ''n''−1
| |
| :'''do''' ''d''<sub>''i''+1</sub> ← random element of { 0, ..., ''i'' }
| |
| :: ''a''[''i''] ← ''a''[''d''<sub>''i''+1</sub>]
| |
| :: ''a''[''d''<sub>''i''+1</sub>] ← ''i''
| |
| </blockquote>
| |
| | |
| If ''d''<sub>''i''+1</sub> = ''i'', the first assignment will copy an uninitialized value, but the second will overwrite it with the correct value ''i''.
| |
| | |
| ====Generation in lexicographic order====
| |
| | |
| There are many ways to systematically generate all permutations of a given sequence.<ref name=sedegewick1977>{{cite journal
| |
| |last=Sedgewick|first=R
| |
| |title=Permutation generation methods
| |
| |journal=Computing Surveys|year=1977|volume=9
| |
| |pages=137–164
| |
| |url=http://www.math.uiowa.edu/~goodman/22m150.dir/2007/Permutation%20Generation%20Methods.pdf
| |
| }}</ref>
| |
| One classical algorithm, which is both simple and flexible, is based on finding the next permutation in [[lexicographic ordering]], if it exists. It can handle repeated values, for which case it generates the distinct multiset permutations each once. Even for ordinary permutations it is significantly more efficient than generating values for the Lehmer code in lexicographic order (possibly using the [[factorial number system]]) and converting those to permutations. To use it, one starts by sorting the sequence in (weakly) [[increasing]] order (which gives its lexicographically minimal permutation), and then repeats advancing to the next permutation as long as one is found. The method goes back to [[Narayana Pandit]]a in 14th century India, and has been frequently rediscovered ever since.<ref name="Knuth fasc2">{{Cite book
| |
| | last=Knuth | first = D. E. | author-link = Donald Ervin Knuth
| |
| | title = The Art of Computer Programming
| |
| | volume =4, Fascicle 2
| |
| | contribution = Generating All Tuples and Permutations | publisher = Addison-Wesley | year = 2005
| |
| | isbn = 0-201-85393-0
| |
| | pages = 1–26}}</ref>
| |
| | |
| The following algorithm generates the next permutation lexicographically after a given permutation. It changes the given permutation in-place.
| |
| | |
| # Find the largest index ''k'' such that {{nowrap|''a''[''k''] < ''a''[''k'' + 1]}}. If no such index exists, the permutation is the last permutation.
| |
| # Find the largest index ''l'' such that {{nowrap|''a''[''k''] < ''a''[''l'']}}.
| |
| # Swap the value of ''a''[''k''] with that of ''a''[''l''].
| |
| # Reverse the sequence from ''a''[{{nowrap|''k'' + 1}}] up to and including the final element ''a''[''n''].
| |
| For example, given the sequence [1, 2, 3, 4] which starts in a weakly increasing order, and given that the index is [[Zero-based numbering|zero-based]], the steps are as follows:
| |
| # Index ''k'' = 2, because 3 is placed at an index that satisfies condition of being the largest index that is still less than ''a''[''k'' + 1] which is 4.
| |
| # Index ''l'' = 3, because 4 is the only value in the sequence that is greater than 3 in order to satisfy the condition ''a''[''k''] < ''a''[''l''].
| |
| # The values of ''a''[2] and ''a''[3] are swapped to form the new sequence [1,2,4,3].
| |
| # The sequence after ''k''-index ''a''[2] to the final element is reversed. Because only one value lies after this index (the 3), the sequence remains unchanged in this instance. Thus the lexicographic successor of the initial state is permuted: [1,2,4,3].
| |
| Following this algorithm, the next lexicographic permutation will be [1,3,2,4], and the 24th permutation will be [4,3,2,1] at which point ''a''[''k''] < ''a''[''k'' + 1] does not exist, indicating that this is the last permutation.
| |
| | |
| ====Generation with minimal changes====
| |
| {{main|Steinhaus–Johnson–Trotter algorithm}}
| |
| {{main|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.<ref name="Knuth fasc2"/>
| |
| An old paper<ref>{{cite journal|last=Heap|first=B. R.|title=Permutations by Interchanges|journal=The Computer Journal|year=1963|volume=6|issue=3|pages=293–4|url=http://comjnl.oxfordjournals.org/content/6/3/293.full.pdf}}</ref> proposed a very efficient algorithm called [[Heap's algorithm|Heap's permutation generation algorithm]]. An article<ref name=sedegewick1977 /> authored by [[Robert_Sedgewick_(computer_scientist)|Robert Sedgewick]] said that this algorithm is the fastest algorithm of generating permutations in applications.
| |
| | |
| ====Meandric Permutations====
| |
| An extension of this algorithm was introduced in 2011 for the generation of [[meandric number]] order n. In this case all the numbers that begin with the digit 1 and have unique alternate even-odd digits can be exactly generated avoiding the commonly used (2n)! permutations of the first n components, before any further classification or proof of the meanders' type. <ref>{{cite journal
| |
| |last1=Alexiou |first1=A. |last2=Psiha |first2=M. |last3=Vlamos |first2=P.
| |
| |title= Combinatorial permutation based algorithm for representation of closed RNA secondary structures
| |
| |journal=Bioinformation|year=2011|volume=7(2)
| |
| |pages=91–95
| |
| |url= http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3174042/
| |
| }}</ref>
| |
| | |
| Example of M3 Permutations
| |
| <br />
| |
| {| class="wikitable"
| |
| |-
| |
| ! Initial Sets !! Transposition of digits !! Meandric Permutations
| |
| |-
| |
| | 1-2-3-4-5-6 || || 1-2-3-4-5-6
| |
| |-
| |
| | 1-2-3-4-5-6 || 4,6 || 1-2-3-6-5-4
| |
| |-
| |
| | 1-2-3-4-5-6 || 2,6 || 1-6-3-4-5-2
| |
| |-
| |
| | 1-2-5-4-3-6 || || 1-2-5-4-3-6
| |
| |-
| |
| | 1-2-5-4-3-6 || 4,6 || 1-2-5-6-3-4
| |
| |-
| |
| | 1-2-5-4-3-6 || 2,6 || 1-6-5-4-3-2
| |
| |-
| |
| | 1-4-3-2-5-6 || || 1-4-3-2-5-6
| |
| |-
| |
| | 1-4-3-2-5-6 || 2,6 || 1-4-3-6-5-2
| |
| |-
| |
| | 1-4-3-2-5-6 || 4,6 || 1-6-3-2-5-4
| |
| |-
| |
| | 1-4-5-2-3-6 || || 1-4-5-2-3-6
| |
| |-
| |
| | 1-4-5-2-3-6 || 2,6 || 1-4-5-6-3-2
| |
| |-
| |
| | 1-4-5-2-3-6 || 4,6 || 1-6-5-2-3-4
| |
| |}
| |
| | |
| ===Software implementations===
| |
| | |
| ====Calculator functions====
| |
| Many [[scientific calculator]]s and computing software have a built-in function for calculating the number of ''k''-permutations of ''n''.
| |
| | |
| * Casio and TI calculators: '''nPr'''
| |
| * HP calculators: '''PERM'''<ref>http://h20331.www2.hp.com/Hpsub/downloads/50gProbability-Rearranging_items.pdf</ref>
| |
| * Mathematica: '''FallingFactorial'''
| |
| | |
| ====Spreadsheet functions====
| |
| Most [[Spreadsheet|spreadsheet software]] also provides a built-in function for calculating the number of ''k''-permutations of ''n'', called PERMUT in many popular spreadsheets.
| |
| | |
| ===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 <ref>[http://www.3gpp.org/ftp/Specs/html-info/36212.htm 3GPP TS 36.212]</ref>).
| |
| Such applications raise the question of fast generation of permutations satisfying certain desirable properties. One of the methods is based on the [[permutation polynomials]].
| |
| | |
| ==See also==
| |
| {{Portal|Mathematics}}
| |
| {{Div col}}
| |
| *[[Alternating permutation]]
| |
| *[[Binomial coefficient]]
| |
| *[[Combination]]
| |
| *[[Combinatorics]]
| |
| *[[Convolution]]
| |
| *[[Cyclic order]]
| |
| *[[Cyclic permutation]]
| |
| *[[Even and odd permutations]]
| |
| *[[Factorial number system]]
| |
| *[[Josephus permutation]]
| |
| *[[Levi-Civita symbol]]
| |
| *[[List of permutation topics]]
| |
| *[[Major index]]
| |
| *[[Necklace (combinatorics)]]
| |
| *[[Permutation group]]
| |
| *[[Permutation pattern]]
| |
| *[[Permutation polynomial]]
| |
| *[[Permutation representation (symmetric group)]]
| |
| *[[Probability]]
| |
| *[[Random permutation]]
| |
| *[[Rencontres numbers]]
| |
| *[[Sorting network]]
| |
| *[[Substitution cipher]]
| |
| *[[Superpattern]]
| |
| *[[Symmetric group]]
| |
| *[[Twelvefold way]]
| |
| *[[Weak order of permutations]]
| |
| {{Div col end}}
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| * [[Miklós Bóna]]. "Combinatorics of Permutations", Chapman Hall-CRC, 2004. ISBN 1-58488-434-7.
| |
| * [[Donald Knuth]]. ''The Art of Computer Programming'', Volume 4: ''Generating All Tuples and Permutations'', Fascicle 2, first printing. Addison–Wesley, 2005. ISBN 0-201-85393-0.
| |
| * Donald Knuth. ''The Art of Computer Programming'', Volume 3: ''Sorting and Searching'', Second Edition. Addison–Wesley, 1998. ISBN 0-201-89685-0. Section 5.1: Combinatorial Properties of Permutations, pp. 11–72.
| |
| * Humphreys, J. F.. ''A course in group theory''. Oxford University Press, 1996. ISBN 978-0-19-853459-4
| |
| | |
| ==External links==
| |
| * {{springer|title=Permutation|id=p/p072270}}
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| | |
| {{Commons category|Permutations}}
| |
| | |
| [[Category:Abstract algebra]]
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| [[Category:Combinatorics]]
| |
| [[Category:Factorial and binomial topics| ]]
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| [[Category:Permutations| ]]
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Louis Vuitton København fordi det gav os noget
Ikke fortalte mig, men hvad han bad mig om at gøre, 'Arthur said.Prosecutors sagde, at han var et afgørende vidne.' Selvfølgelig tror jeg Arthur marts vidneudsagn var afgørende, fordi det gav os noget, vi ikke har, og det var mindst en forklaring på, hvad der skete med Janet marts krop, 'Anklageren i Perry marts sagen Tom Thurman said.In kendsgerning, efter at have hørt Arthur vidner, juryen dømt Perry i mord.' Han bad Louis Vuitton København mig om at hjælpe afhænde Janet krop, 'Arthur sagde i sin vidneforklaring.
Under kraftig regn, vand, der strømmer ind i vådområder breder sig og naturligt mister sin hastighed, så miljøet til naturligt at filtrere Ray Ban Briller giftstoffer (EPA, 2010). Fjerde udvide dig selv! Jeg nåede så højt som jeg kunne med min højre hånd, før du slipper bolden.
Jeg ser en anden ob praksis denne gang (flyttet fra ud af staten), men de fortsætter med at give mig de store baby-bs. Denne undersøgelse viste, at kun 27,5% af patienterne havde deres urinsyre niveau målt på mindst årsbasis. Denne blodprop kan føre til hjernen hævelse eller et slagtilfælde, og det er bestemt livstruende.
Undskyld mig, men jeg iagttog BBC produktion under krigen, og det var ofte chokerende at se netværkets angiveligt uvildige journalister og ankre hengive sig i spydige sidebemærkninger om amerikanerne eller præsident Bush, som spørger, der afslørede en anti Polo Ralph Lauren Outlet krig og anti amerikansk dagsorden, for ikke at nævne den enorme spørgsmål om at jagte Nike Roshe Tony Blair rundt for forbrydelsen være så venlige til amerikanerne..
2 år fra da hun døde af kræft. For eksempel kan du vide, hvornår din søster ligger fordi hun hæver hendes øjenbryn. Modificeret skrev: Jeg er stedet hurtigt i mit liv. Art Institute Bygningen har den usædvanlige egenskab af fælles open air jernbanespor.
Niveauer blev vurderet ved baseline på fem og 15 uger og seks, ni, 12, 18 og 24 måneder efter start af behandling.. Jeg tror han er på vej til at blive en speciel spiller.. Charteret blev vedtaget på en anden Haagerkonferencen den 20. Selvom Halliburton forudsat data fra en af prøverne til BP i en 8 Marts e-mail, efterforskerne sagde, var der ingen tegn på, at Halliburton fremhævede betydningen af denne information eller at BP arbejdere rejst spørgsmål om det..
Hele kæden skal arbejde sammen. Fra min erfaring, de gør en webside og sidde der. Id gå se på, hvad der tilbydes nu, så du har en idé. Se det i Twitter og jeg ser det i tekstbeskeder. Det kom fra det gamle spanske ord Camarada, hvilket betyder meget det samme som den franske version, men især i forbindelse med kammerater i en kaserne.
Ray Ban Clubmaster jeg ved
Ja, jeg ved, du tror, du fik det trådet korrekt, men symaskiner er meget specielt om at blive udført korrekt. Det karakteren af dyret.. Selvfølgelig mange af dem, der er farvet i uld Labor Fans har forsøgt at få spillet kommentar på emnet næste sæson spiller, men Abbott står over for et hold, der holder score mål for ham snarere end mod ham, så han ikke har til at Ray Ban Clubmaster arbejde meget hårdt for at få det vindende score. Sports-fans få let kede af et spil, der er for en ensidig og de begynder at glide ud til ølteltet godt før sidste fløjt.
De to vigtigste faktorer for søgemaskine optimering er effektive titler og Nike Roshe Run klar brødtekst. Sørg for, at titlerne er relevante, unik for hver side, og omfatter centrale sætninger, der sandsynligvis skal søges. Når maden er knap, din krop går i sult-tilstand, sende næringsstoffer til vigtige organer som hjerte og hjerne og springe din sultne hårsække. Det er derfor, kroniske Dieters har ofte lank Lokker..
Vi har købt en masse spillere denne sommer, og det er nu at ligne hans team. Det rimeligt at sige, at han ikke er at hjælpe sig selv med nogle af beslutningsprocessen. (Faktisk en General som Beck og en Admiral ligesom Canaris ønskede at fjerne eller dræbe Hitler, men ophørte indsats og planer som manden overtog Kontrol med Chamberlain eftergivenhed.) I lighed med denne periode i 30'erne, var der mere end ufuldbyrdede vold i slutningen af 60'erne i Bundesrepublik. Ekstra parlamentary strejker, protester og, i nogle tilfælde dødelig vold.
De dart over julen er nu en stor tradition blandt fans, og næsten hver session blev udsolgt på Alexandra Palace, som mere end 50.000 fans deltog i begivenheden fra første runde til finalen. Som altid, fancy kjole var dagens orden, og der var masser af underlige og vidunderlige kostumer til at nyde, sammen med masser af sange og drilleri.
Det er interessant, at man som lærer, da jeg stødte på et problem så grundlæggende som at lære eleverne at afkode Jeg henvendte mig til kilder uden for, hvad jeg har lært i college. Det er også vigtigt at bemærke, at jeg er uddannet i foråret 2001. Jeito e raciocnio. Eu mesmo testei en prova e estava tudo certo' insistia ele..
Raikkonen toer sidste søndag efter at have Ray Ban Solbriller bestået Grosjean i mellem Safety Car re starter ved Yeongam. Finnen kapitaliseres på en fejl ved at holdkammeraten at overhale i sving 1, men Grosjean, stadig at tro sig at være hurtigere og derfor i stand til at montere en stærkere udfordring til vinderen Sebastian Vettel, bad efterfølgende holdet at skifte Ray Ban Briller ordren..
Michael Kors Tasker ligesom Facebooks
Familier er velkomne. Michael Kors Tasker Voksne skal ledsage børnene og betale programmet gebyr. 4C s Change EverythingTwitter børsnotering, ligesom Facebooks, LinkedIn og de mange andre værktøjer, som for nylig er blevet skabt for at ændre den måde, vi forbinder, kommunikere, samarbejde og skabe og forme fællesskaber (4C s) bringer Nike Air Force One bæredygtighed til disse teknologier. Ikke bare et modefænomen (hvilket er præcis, hvad folk kaldte fastnet telefon over 130 år siden) en børsintroduktion viser, at markedet validerer iboende værdi inden for værktøj til at forbedre på 4C er.
Januar) hævder, at den hvide ting ikke kun forårsager fedme, men er også knyttet til andre alvorlige sundhedsmæssige forhold, herunder dårlig hjernens udvikling hos børn, grå stær og endda Alzheimers. Den kost, der er udviklet af en ernæringsekspert og hudlæge, indebærer en tre-dages detox at befri dit system af sukker og en tre-dages hud fix til at hjælpe dig se og føle stor.
Network discovery Beats By Dre Studio er på i private netværk og fra i offentlige. File (mappe) deling giver indstillingerne for den delte data, du har oprettet Polo Ralph Lauren Outlet dig selv. Kotik Cannabis gratis multimedieklip, sort ikon skitsere symbol tegning planter blad cannabis tegneserie logoer gratis potteplante draw logo ukrudt tatovering Kotik designs tegninger. Indsamling af cool computer tekst symboler og tegn på, at du kan bruge på Facebook og andre steder.
Disse fokuserer på en formodet ulovlig hack, stjålne e-mails osv. Reaktion blandt de blogs, og især de mere alvorlige af disse (Bishop Hill, WUWT, CA, Lucia, Air Vent, de Pielkes, Philip Stott) har været at forstå alvoren meget af det materiale, der er blevet lækket.
Kvartal i forhold til samme periode sidste år. Licensing salget er steget, opvejet af lavere franchise-provisioner.. Pure svar 'Valley Heights' plys madras er designet til at give dig en god nats søvn, så du kan få mere energi i løbet af dagen. Den er fremstillet med strik stof tikkende, Advanced Comfort Quilt, FireBlocker, Convoluted Skum, en halv tomme af quiltet latex-skum, 5 inches af Talalay latex-skum, Comfort Skum og Serta Foam Core..
Jeg har en familie på 4 + min ven, så der er 5 Jeg fodring. I aften har jeg lavet en Gumbo ud af det, og vi alle elskede det. Siden starten af kampagnen i 2008, har Weight Watchers doneret 5500 tusind dollars til sine to partnerorganisationer sult kæmpende organisationer. De midler doneret til No Kid Hungry kampagne hjælpe bestræbelser på at bringe barndom sult i Amerika ved at forbinde børn i nød med sund mad, hvor de bor, lære og spille, mens bidrag til bekæmpelse af sult fortsætter til brændstof organisationens bestræbelser på at udrydde barndommen fejlernæring i globalt.
Michael Kors Danmark initiativrig
Jeg ville temmelig meget garanti for, at hver eneste af jer derude er en patriot. Vinderen var en pige, og piger deltog i 50% af de sjove campister i år! Med mulighed for børnene at Michael Kors Danmark være en pilot i en flyvemaskine, vi har sat grundlaget for fremtidige luftfart karriere..
For at få alle de essentielle næringsstoffer, du har brug for 8 timers Powerfoods. Normalt dette interval er mellem tre og seks måneder, men det kan endda være otte måneder til et år.. Så Ralph Lauren Danmark trække på dine læder jakker og hoved downtown eller hvor søjlerne er i din by og bestille en drink ved hver stop, indtil du har nået din grænse.
Grubb, søn af Grubb teleskop bygning familie i Dublin, designet observatorium og byggede de astronomiske instrumenter til strukturen. Og en eller anden måde bare at vide, at andre forældre har gjort langt værre ting er balsam for den pågældende forældre skyld.
Overvåget fleksible teams af lokalsamfundet moderatorer og support personale for at styrke medlem oplevelse. Du vil have det til at reflektere over, hvad din gruppe står for, Abercrombie And Fitch Dk og hvordan du gør en forskel i dit samfund. Var sådan en stærk måned at December bare kan være det faktum, at en stor del af væksten skete tidligere på året i modsætning til senere, sagde Sadiq Adatia, chef investering officer på Sun Life Financial.
Jeg bare se bolden, og jeg er ligesom, 'Nå, er du nødt til at gøre det', og derefter dykke jeg. Tænk på dem som 'cooldown' rum, hvor din familie kan tage en pause. Det firma, der hyrede mig var at lukke omkring en ud af seks af disse præsentationer, men ønskede at øge deres antal.
Vi har samlet et computere svarene på de kategorier folk spørger mest om West Nile virus og West Nile-virus Ray Ban Briller markedsføring. Nogle mennesker kan lide at opleve hver eneste tomme af deres rejse og se forskellige steder undervejs. At gøre landlording en lettere opgave, skal du være wellequippedto håndtere de problemer, du vil møde.
Du kan først lære de ins og outs af online-salg på nogen af disse steder, da de er et væld af oplysninger, og så kan du gå på at åbne din egen online-butik.. Kun børn tendens til at være ambitiøs, initiativrig, energisk og villig til at yde ofre for at blive en succes, skriver Leman.
Nu vil han ikke noget at gøre med booster enten.. Ret. Den eneste virkelige big deal IMO er carb vil sandsynligvis nødt til at blive genopbygget. Afrikanske mæglere mødtes med repræsentanter for de rivaliserende sider repræsenterer præsident Salva Kiir og fortrængte vicepræsident Riek Machar, briefing dem separat forud for den officielle start på direkte forhandlinger.
Beats By Dre Studio som vil ændre den måde
Da jeg fandt Hverdagens velsignelser jublede jeg. Rettet mod enhver form for dyr, herunder hunde og mennesker, eller selv livløse genstande. Hun var altid bare dette endnu hærdet person. Han afviser øko ideer rask, én efter én. Programmet hævder at gøre én ting, som foregiver at være et spil, men i Beats By Dre Studio stedet gøre noget andet, når du kører dem.
Folk har sovet på barnesenge i den forløbne uge, tre meter fra hinanden i en hockey arena, sagde han. Selvfølgelig er der nogle aspekter af bouillabaisse af ændringer, som Labor uvederhæftigt har rullet ind i Carbon skat regninger, ligesom Ralph Lauren Danmark at hæve skattefri tærskel, som er noget længst, men jeg don tror, at de vil blive sødning politisk cyanid nok til at redde dette land ældste politiske parti fra glemsel ved næste valg.
Jeg har altid, altid undre sig, når du læser om herskabet, de betaler for dette? Hvem gør vasketøjet? Longbourn henvender det andet spørgsmål i stor detalje, helt ned hvordan skørter fik farves, hvordan vaskeri sæbe er madeandhow pigen skincracked og blødte fra chilblains.
Sætte en til salg skilt på forsiden græsplænen eller har sheriffen sætte meddelelsen ejectment på døren har også været rigtig stærk i at få en låntager til endelig at forstå, at det første, vi kan afskærme (ja, selv fra den 2. Her er hans 10 måder at gå videre end opmærksomhed: Gift og RewardsHumanization af processBuild en storyTo føle eller til diePlease investere 22 minutter til at se Brian stor presentationand få et dybere indblik i hans advice.The hotteste tech companiesRobert Scoble tog ordet og i omkring 20 minutter han gennemgik de hotteste kommende apps og virksomheder, som vil ændre den måde, vi interagerer.
Der er mange flere mennesker, der nu er ramt af dine beslutninger. Dette er alle gjort i Irland.' Den nye chip er designet til at blive brugt til magten Michael Kors København dagligdags emner, fra 'wearable' teknologi (såsom smartwatches og briller) til klimaanlæg og flåde bil management.
Jeg har også set på ejendomme i Port St. Børnene blev let distraheret, og ved udgangen af masse jeg havde dem sidder i hjørnet! Med Beats By Dre stress og jag i julen, er det nemt at glemme 'årsagen til sæsonen'. Det medfører også inflammation og et fald i styrke og magt, så kroppen kan helbrede og gøre dig stærkere.
Den vil undersøge paralleller mellem den afghanske oprør mod det britiske imperium og den aktuelle opstand mod de NATO-styrker i det betrængte land. Hun forlod aldrig hans side. Dette er også en meget god ting. Eksponering af hjernen til testosteron i livmoderen kan ændre spædbarn adfærd: unge mænd viser mere aggressiv som en leg end kvinder.