Incomplete date November 2011 A randompermutation is a random ordering of a set of objects, that is, a permutation valued random variable . The use of random permutations is often fundamental to fields ... of a randompermutation is the shuffling of a card deck deck of cards this is ideally a randompermutation of the 52 cards. Generating random permutations Entry by entry brute force One method of generating a randompermutation of a set of length n uniform distribution discrete uniformly at random i.e., each of the n permutation s is equally likely to appear is to generate a sequence by taking a random number between 1 and n sequentially, ensuring that there is no repetition, and interpreting this sequence x sub 1 sub , ..., x sub n sub as the permutation math begin pmatrix 1 & 2 & 3 & cdots ... distributed randompermutation approaches a Poisson distribution with expected value 1 as n grows ... genetics Golomb Dickman constant Perfect shuffle Randompermutation statistics Shuffle Shuffling ... http mathworld.wolfram.com RandomPermutation.html Randompermutation at MathWorld http www.techuser.net randpermgen.html Randompermutation generation detailed and practical explanation of Knuth shuffle ... retries whenever the random number picked is a repeat of a number already selected. This can ... , one chooses a number j at random between 1 and n &minus i 1 and sets x sub i sub equal to the j th largest of the unchosen numbers. Knuth shuffles A simple algorithm to generate a permutation of n items uniformly at random without retries, known as the Knuth shuffle , is to start with any permutation for example, the Identity function identity permutation , and then go through the positions ... element from positions i through n , inclusive. It s easy to verify that any permutation of n elements ... over all such permutations. Statistics on random permutations Fixed points main Rencontres ... random processes, the quality of the resulting distribution of an implementation of a randomized algorithm ... more details
The statistics of random permutations , such as the Permutation group Examples cycle structure of a randompermutation are of fundamental importance in the analysis of algorithms , especially of sorting ... a cousin of quicksort to select a random element of a randompermutation. Quickselect will perform ... functions of randompermutation statistics. Hence it is of vital importance to compute these generating functions. The article on randompermutation s contains an introduction to random ... that a randompermutation is an involution. Number of permutations that are m th roots of unity This generalizes ... and the probability that a randompermutation is a derangement is math 1 e. math This result may ... may agree in advance on a randompermutation of their names. We consider the general ... Mathematiques.net . Consult the Randompermutation statistics 100 prisoners references on 100 prisoners ... of cycles of size m in a permutation of length n less than m is zero obviously . A randompermutation of length at least m contains on average 1 m cycles of length m . In particular, a randompermutation ... number of cycles of length at most m in a randompermutation is about ln m . Moments of fixed ... be the number of fixed points of a randompermutation. Using Stirling number Stirling number of the second ... k end matrix right . math Expected number of cycles of any length of a randompermutation We construct ... math log n math . Expected number of transpositions of a randompermutation We can use the disjoint ... element q of a randompermutation math sigma math and ask about the expected size of the cycle that contains ... a random element of math n math of a randompermutation, the element lies on a cycle of size m ... containing m elements and a randompermutation, and ask about the probability that all elements of Q ... see Randompermutation statistics External links external links . The term odd cycle invariant ... competition A link to the Putnam competition website appears in the section Randompermutation statistics ... more details
permutation of n to the sequence, or choose a random element from the set of distinct multiset permutations ... for random generation. The basic idea to generate a randompermutation is to generate at random one ... a randompermutation of a 0 , a 1 , ..., a n 1 can be described as follows in pseudocode blockquote ... Permutation pattern Permutation polynomial Probability Randompermutation Rencontres numbers Sorting ... of permutation is used with several slightly different meanings, all related to the act of permuting rearranging objects or values. Informally, a permutation of a set of objects is an arrangement ... as a permutation of its letters. The study of permutations in this sense generally belongs to the field ... of sorting algorithm s in computer science. In algebra and particularly in group theory , a permutation ... a problem requires studying certain permutations related to it. Generalities The notion of permutation ... 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 ... of multisets In combinatorics , a permutation is usually understood to be a sequence containing each ... a permutation of a finite set S of n elements is equivalent to a bijection from 1, 2, ... , n to S in which ... n permutations of S . There is also a weaker meaning of the term permutation that is sometimes ..., more common, meanings of permutation . The number of such k permutations of n is denoted ... of n . If M is a finite multiset , then a multiset permutation is a sequence of elements of M in which ..., the term permutation of a set means a bijective map, or bijection , from that set onto itself. The set ... 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 ... more details
In cryptography , the term pseudorandom permutation , abbreviated PRP , refers to a function that cannot be distinguished from a randompermutation that is, a permutation selected at random with uniform probability, from the family of all permutations on the function s domain with practical effort. A pseudorandom permutation family is a collection of pseudorandom permutations, where a specific permutation may be chosen using a key. The idealized abstraction of a block cipher is a truly randompermutation. If a distinguishing algorithm exists that achieves significant advantage cryptography advantage with less effort than specified by the block cipher s security parameter this usually means the effort required should be about the same as a brute force search through the cipher s key space , then the cipher is considered broken at least in a certificational sense, even if such a break doesn t immediately lead to a practical security failure. See also Block cipher pseudorandom permutation families operating on fixed size blocks of bits Format Preserving Encryption pseudorandom permutation families operating on arbitrary finite sets Pseudorandom function References cite book author Mihir Bellare , Phillip Rogaway date 2005 09 20 title Introduction to Modern Cryptography url http www cse.ucsd.edu mihir cse207 classnotes.html chapter Chapter 3 Pseudorandom functions accessdate 2007 09 30 Category Theory of cryptography Category Cryptographic primitives crypto stub ... more details
wiktionary permutationPermutation is a mathematical concept. Permutation may also refer to Permutation music , a concept in musical set theory Permutation policy debate , a type of argument in policy debate Permutation album Permutation album , an album by Brazilian electronic artist Amon Tobin Permutation , an instrumental song by the Red Hot Chili Peppers Resampling statistics Permutation tests Permutation test , in statistics disambig sk Permut cia ... more details
unreferenced date June 2009 In classical cryptography , a permutation cipher is a transposition cipher in which the cryptographic key key is a permutation . To apply a cipher, a randompermutation of size e is generated the larger the value of e the more secure the cipher . The plaintext is then broken into segments of size e and the letters within that segment are permuted according to this key. In theory, any transposition cipher can be viewed as a permutation cipher where e is equal to the length of the plaintext this is too cumbersome a generalisation to use in actual practice, however. Identifying the cipher Because the cipher doesn t change any of the characters, the ciphertext will have exactly the same letter frequencies as the underlying plaintext. This means that the cipher can in many cases be identified as a transposition by the close similarity of its letter statistics with the letter frequencies of the underlying language. Breaking the cipher Because the cipher operates on blocks of size e , the plaintext and the ciphertext have to have a length which is some multiple of e . This causes two weaknesses in the system first, the plaintext may have to be padded if the padding is identifiable then part of the key is revealed and second, information relating to the length of the key is revealed by the length of the ciphertext. To see this, note that if the ciphertext is of length i then e must be one of the divisor s of i . With the different possible key sizes different possible permutations are tried to find the permutation which results in the highest number of frequent bigram s and trigram s as found in the underlying language of the plaintext. Trying to find this permutation is essentially the same problem encountered when analysing a Transposition cipher Columnar transposition columnar transposition cipher multiple anagram ming . See also Topics in cryptography Cryptography navbox classical Category Classical ciphers Category Permutations hu Permut ci s ... more details
Multiple issues orphan March 2012 confusing June 2010 unreferenced May 2010 In combinatorics combinatorial mathematics , given a set S and two subsets U and V , a bijection from U to V is a partial permutation of S . Thus any permutation is a partial permutation with U V . Another way of looking at it is that a partial permutation on S is a partial function on S which can be extended to a permutation of S . Category Combinatorics Category Functions and mappings combin stub ... more details
Image Permutation graph.svg thumb 300px The permutation 4,3,5,1,2 and the corresponding permutation graph In areas of mathematics influenced by graph theory , a permutation graph is the intersection graph ... plane . Equivalently, given a permutation &sigma sub 1 sub ,&sigma sub 2 sub ,&sigma sub 3 sub ,... of the numbers 1,2,3,... n , a permutation graph has a vertex for each number 1,2,3,... n and an edge between any two numbers that are in reversed order in the permutation i.e. an edge between any two numbers where the segments cross in the permutation diagram. A permutation graph has a unique representation as a permutation diagram if and only if it is prime with respect to the modular decomposition ... is a permutation graph if and only if G is a circle graph that admits an equator, i.e., an additional ... 4.7.1, p.57. ref A graph G is a permutation graph if and only if both G and its complement graph complement ... G is a permutation graph if and only if it is the comparability graph of a partially ordered set that has order dimension at most two. ref harvtxt Baker Fishburn Roberts 1971 . ref If a graph G is a permutation graph, so is its complement. A permutation that represents the complement of G may be obtained by reversing the permutation representing G . Efficient algorithms As a subclass of the perfect graph s, many problems that are NP complete for arbitrary graphs may be solved efficiently for permutation graphs. For instance the largest clique graph theory clique in a permutation graph corresponds to the longest increasing subsequence longest decreasing subsequence in the permutation defining the graph, so the clique problem may be solved in polynomial time for permutation graphs by using ... subsequence in a permutation corresponds to an independent set of the same size in the corresponding permutation graph. the treewidth and pathwidth of permutation graphs can be computed in polynomial ... minimal vertex separators in a permutation graph is polynomial in the size of the graph. ref harvtxt ... more details
technical date November 2011 In combinatorics combinatorial mathematics , an unpredictable permutation UP F sub k sub is a permutation where a probabilistic polynomial time PPT adversary cannot predict the outcome of the challenge query F sub k sub , that was queried on a new input m during the challenge round. This happens after the adversary queries F sub k sub on many inputs before finally accepting the challenge. Here, the adversary has access to an oracle for both forward and inverse permutation operations and has to predict an unqueried input output pair. This is the challenge 1 1 3 3 . It can be shown that a function F sub k sub is not a secure message authentication code MAC if it satisfies only the unpredictability requirement 1 1 . It can also be shown that one cannot build an efficient variable input length MAC from a block cipher which is modelled as an UP of n bits. It has been shown that the output of a k n log round Feistel construction with unpredictable round functions may leak all the intermediate round values. Even for realistic Unpredictable Functions UF , some partial information about the intermediate round values may be leaked through the output. It was later shown that if a super logarithmic number of rounds in the Feistel construction is used, then the resulting UP construction is secure even if the adversary gets all the intermediate round values along with the permutation output 4 4 . There is also a theorem that has been proven in this regard which states that if there exists an efficient UP adversary A sub sub that has non negligible advantage sub sub in the unpredictability game against UP construction sub U,k sub and which makes a polynomial number of queries to the challenger, then there also exists a UF adversary ... being substituted with weaker unpredictability. Verifiable unpredictable permutations are the permutation ... Naor, International Association for Cryptologic Research. div div id 5 5. Verifiable Random Functions ... more details
Unreferenced date April 2011 A cyclic permutation or circular permutation is a permutation built from one or more Set mathematics sets of elements in cyclic order . The notion cyclic permutation is used in different, but related ways Definition 1 image 050712 perm 1.png right mapping of permutation A permutation P over a Set mathematics set S with k elements is called a cyclic permutation with offset t if and only if the elements of S may be total order ordered c 1 c 2 ... c k and the mapping of P ... cyclic permutation of definition type 1 will be constructed with exactly greatest common divisor gcd ... & 6 & 8 3 & 4 & 5 & 7 & 6 & 8 & 1 & 2 end pmatrix 1356 2478 math is a cyclic permutation with offset ... i else. Definition 2 image 050712 perm 2.png right mapping of permutation A permutation is called a cyclic permutation if and only if it will be constructed with exactly 1 cycle. Note Every permutation over a set with k elements is a cyclic permutation of definition type 2 if and only if it is a cyclic permutation of definition type 1 with gcd k , offset 1 Example math begin pmatrix 1 & 2 & 3 ... right mapping of permutation A permutation is called a cyclic permutation if and only if only one of the constructing cycles will have length 1. Note Every cyclic permutation of definition type 3 may be seen as an union mathematics union of a cyclic permutation of definition type 2 and some fixed point mathematics fixed points . Every cyclic permutation of definition type 2 may be seen as a cyclic permutation of definition type 3 with zero fixed points. Example math begin pmatrix 1 & 2 & 3 & 4 ... 4 & 6 & 8 & 3 & 7 & 1 & 2 & 5 end pmatrix 146837 2 5 math See also Cyclic permutation of integer Cycle notation Cycles and fixed points Stirling number Caesar cipher Circular permutation in proteins Category Abstract algebra Category Permutations ar eo Cikla permuto fr Permutation circulaire mk pt Permuta o circular sr sv Cyklisk permutation ... more details
Groups In mathematics , a permutation group is a group mathematics group G whose elements are permutation ... permutation group is usually restricted to mean a subgroup of the symmetric group. The symmetric ... permutations of M is often written as Sym M . The application of a permutation group to the elements ... As a subgroup of a symmetric group, all that is necessary for a permutation group to satisfy the group mathematics group axioms is that it contain the identity permutation, the inverse permutation of each permutation it contains, and be closed under function composition composition of its permutations ... M 1,2,3,4 , a permutation g of M with g 1 2, g 2 4, g 4 1 and g 3 3 will be written as 1,2,4 ... of Groups to Solve Rubik s Cube ref This is the identity, the trivial permutation which fixes each element. a 1 2 3 4 1 2 This permutation interchanges 1 and 2, and fixes 3 and 4. b 1 2 3 4 3 4 Like the previous one, but exchanging 3 and 4, and fixing the others. ab 1 2 3 4 This permutation ... forms a group, since aa bb e , ba ab , and baba e . So G , M forms a permutation group. The Rubik s Cube puzzle is another example of a permutation group. The underlying set being permuted is the coloured subcubes of the whole cube. Each of the rotations of the faces of the cube is a permutation of the positions .... More generally, every group G is isomorphic to a subgroup of a permutation group by virtue of its ... If G and H are two permutation groups on the same set X , then we say that G and H are isomorphism isomorphic as permutation groups if there exists a bijective map f X X such that r mapsto f ... group isomorphism isomorphic as groups . Notice that different permutation groups may well be isomorphic as abstract groups, but not as permutation groups. For instance, the permutation group on 1,2,3,4 described above is isomorphic as a group but not as a permutation group to 1 2 3 4 , 12 34 ... 1 . For a permutation p in S sub n sub , a pair i , j I sub n sub is a Inversion discrete ... more details
In combinatorics combinatorial mathematics, a separable permutation is a permutation that can be obtained from the trivial permutation 1 by Direct sum of permutations direct sums and Skew sum of permutations skew sums . Separable permutations can also be characterized as the permutations that permutation pattern contain neither 2413 nor 3142. They are enumerated by the Schr der number Schr der numbers . Separable permutations first arose in the work of harvtxt Avis Newborn 1981 , who showed that they are precisely the permutations which can be sorted by an arbitrary number of Stack data structure stacks in series. harvtxt Shapiro Stephens 1991 showed that the permutation matrix of fills up under bootstrap percolation if and only if is separable. The term separable permutation was introduced later by harvtxt Bose Buss Lubiw 1998 . Separable permutations are the permutation analogues of Cograph complement reducible graphs and series parallel partial order s. References Citation last1 Avis first1 David author1 link David Avis last2 Newborn first2 Monroe title On pop stacks in series id MathSciNet id 0624050 year 1981 journal Utilitas Mathematica volume 19 pages 129&ndash 140 . Citation last1 Bose first1 Prosenjit last2 Buss first2 Jonathan last3 Lubiw first3 Anna title Pattern matching for permutations id MathSciNet id 1620935 year 1998 journal Information Processing Letters volume 65 pages 277&ndash 283 doi 10.1016 S0020 0190 97 00209 3 . Citation last1 Shapiro first1 Louis last2 Stephens first2 Arthur B. title Bootstrap percolation, the Schr der numbers, and the N kings problem id MathSciNet id 1093199 year 1991 journal SIAM Journal on Discrete Mathematics volume 4 pages 275&ndash 280 doi 10.1137 0404025 . Category Combinatorics Category Permutations ... more details
elements br The Matrix multiplication product of two permutation matrices is a permutation matrix ... 310px br They are also permutation matrices. In mathematics , in matrix theory , a permutation ... elsewhere. Each such matrix represents a specific permutation of m elements and, when used to multiply another matrix, can produce that permutation in the rows or columns of the other matrix. Definition Given a permutation &pi of m elements, math pi lbrace 1, ldots, m rbrace to lbrace 1, ldots, m ... m end pmatrix , math its permutation matrix is the m × m matrix P sub &pi sub whose entries are all ... permutations &pi and &sigma of m elements and the corresponding permutation matrices P sub &pi sub ..., and of the choice of using the vectors math mathbf e pi i math as rows of the permutation matrix ... , pi math with the permutations in their original order. As permutation matrices are orthogonal ... permutations, there are n permutation matrices. By the formulas above, the n × n permutation ... element . If 1 denotes the identity permutation, then P sub 1 sub is the identity matrix . One can view the permutation matrix of a permutation &sigma as the permutation &sigma of the columns of the identity matrix I , or as the permutation &sigma sup &minus 1 sup of the rows of I . A permutation ... stochastic matrix is a convex combination of permutation matrices of the same order and the permutation ... polytope , the set of doubly stochastic matrices, is the convex hull of the set of permutation matrices. The product PM , premultiplying a matrix M by a permutation matrix P , permutes the rows of M ... algebra trace of a permutation matrix is the number of fixed points of the permutation. If the permutation ... ,..., e sub a sub k sub sub are eigenvector s of the permutation matrix. From group theory we know that any permutation may be written as a product of transposition mathematics transposition s. Therefore, any permutation matrix P factors as a product of row interchanging elementary matrix elementary ... more details
Infobox Album See Wikipedia WikiProject Albums Name Permutation Type Album Artist Amon Tobin Cover Amon Tobin Permutation.jpg Released 1 June 1998 Recorded Genre Electronica Length 69 51 Label Ninja Tune Producer Amon Tobin Reviews Allmusic Rating 4.5 5 Allmusic class album id r352312 pure url yes link Pitchfork Media 9.8 10 http www.pitchforkmedia.com article record review 22602 permutation link Rolling Stone Rating 4 5 http web.archive.org web 20071001211625 http www.rollingstone.com reviews album 300958 permutation 9 Jul 98 Last album Bricolage album Bricolage br 1997 This album Permutation br 1998 Next album Supermodified album Supermodified br 2000 Permutation is the third album by Amon Tobin and second under his own name. It was released in 1998, just over a year after Bricolage album Bricolage . The album was a success for Tobin and found him playing sold out shows at the Montreal Jazz Festival , the Knitting Factory in New York City New York and the Coachella Valley Music and Arts Festival . He went on to release Supermodified album Supermodified in 2000. Track listing Like Regular Chickens 5 16 Bridge 5 56 Reanimator 6 34 Sordid 7 11 Nightlife 6 29 Escape 5 54 Switch 3 49 People Like Frank 6 04 Sultan Drops 5 12 Fast Eddie 7 38 Toys 5 16 Nova 4 42 Notes Amon Tobin makes references to David Lynch films a number of times on Permutation . The song Like Regular Chickens contains a line of dialogue spoken in Eraserhead , while the title of the song People Like Frank is a line of dialogue from Blue Velvet film Blue Velvet . The song also samples a part of Angelo Badalamenti ... track External links http www.ninjatune.net ninja release.php?id 140 Permutation at the Ninja Tune discography features audio clips . MusicBrainz album id c6dcbdf8 5af0 4530 8118 cf6dd68d3659 name Permutation Category 1998 albums Category Amon Tobin albums Category Ninja Tune albums fr Permutation album hu Permutation album nl Permutation pl Permutation ... more details
In combinatorics combinatorial mathematics and theoretical computer science , a permutation pattern is a sub permutation of a longer permutation . The permutation , written as a word in one line notation i.e., in Permutation Notation two line notation with the first line omitted , is said to contain the permutation if there exists a subsequence of entries of that has the same relative order as ... the permutation . The subsequence of need not consist of consecutive entries. For example, permutation ... . Another early landmark result in the field is the Erd s Szekeres theorem in permutation pattern language, the theorem states that for any positive integers a and b every permutation of length at least ... science origins The study of permutation patterns began in earnest with Donald Knuth s consideration of stack data structure stack sorting in CITEREFKnuth1968 1968 . Knuth showed that the permutation ... 1968 , Section 2.2.1, Exercise 13 , which asks how many permutation of n elements are obtainable ... , while Vaughan Pratt showed in CITEREFPratt1973 1973 that the permutation can be sorted by a deque ... k 3,4 k ,1,4 k 2,3, and every permutation that can be obtained from either of these by interchanging ... clear how long it takes to decide if a permutation can be sorted by a deque. harvtxt Rosenstiehl ... by a deque. In his paper, Pratt remarked that this permutation pattern order seems to be the only partial order on permutation that arises in a simple and natural way and concludes by noting that from an abstract point of view , the permutation pattern order is even more interesting than the networks we were characterizing . Enumerative origins main Enumerations of specific permutation classes Another major influence on the early development of the study of permutation patterns came from ... a fixed and typically short permutation. Let Av sub n sub denote the set of permutations of length ..., Simion and Schmidt counted Parity of a permutation even and odd permutations avoiding a pattern of length ... more details
In mathematics , a permutation polynomial for a given finite ring mathematics ring is a polynomial that acts as a permutation of the elements of the ring, i.e. the map math x mapsto g x math is one to one . In case the ring is a finite field , they are under certain assumptions essentially Dickson polynomials ... and correction algorithms. ref name Takeshita1 cite arxiv title Permutation Polynomial Interleavers ... using Permutation Polynomials over Integer Rings year 2005 first1 Oscar last1 Takeshita eprint cs 0506091 ref Quadratic permutation polynomials QPP For the finite ring Z n Z one can construct quadratic permutation polynomials. Actually it is possible if and only if n is divisible by p sup 2 sup for some ... 1 3 g 2 2 g 3 1 math , so the polynomial defines the permutation math begin pmatrix 0 &1 & 2 & 3 0 ... defines the permutation math begin pmatrix 0 &1 & 2 & 3 & 4 & 5 & 6 & 7 0 &3 & 2 & 5 & 4 & 7 & 6 & 1 ... Z . Lemma for k 1 i.e. Z p Z such polynomial defines a permutation only in the case a 0 and b not equal ... defines a permutation if and only if math a 0 mod p math and math b ne 0 mod p math . Rings ... any polynomial math g x a 0 sum 0 i le M a i x i math defines a permutation for the ring Z n Z if and only ... modula math p t k t math . As a corollary one can construct plenty quadratic permutation polynomials ... . Then such polynomial defines a permutation. To see this we observe that for all primes p sub i sub ... and hence is permutation by trivial reason. For the first prime number we should use the lemma discussed previously to see that it defines the permutation. For example, consider Z 12 Z and polynomial math 6x 2 x math . It defines a permutation math begin pmatrix 0 &1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & ... 0 ... g x defines a permutation for the elements of the ring Z p sup k sup Z for k 1 . However ... polynomial math g x ax 3 bx math defines a permutation if and only if for all math y in Z pZ math ... one can see that the analysis of higher degree polynomials to define a permutation is quite subtle ... more details
Refimprove date March 2009 In cryptography , a permutation box or P box is a Methodology method of bit shuffling used to Permutation permute or Transposition cipher transpose bits across Substitution box S boxes inputs, retaining Confusion and diffusion diffusion while Transposition cipher transposing . ref name Cryptography 2007 cite paper author Computer Science and Engineering title Cryptography 2007 url http www.cs.chalmers.se Cs Grundutb Kurser krypto lect03 2x2.pdf booktitle Cryptography 2007 publisher Chalmers University of Technology format Portable Document Format PDF ref In block cipher s, the S boxes and P Boxes are used to make the relation between the plaintext and the ciphertext difficult to understand see Shannon s Confusion and diffusion property of confusion . P boxes are typically classified as compression , expansion , and straight , according as the number of output bits is less than, greater than, or equal to the number of input bits. Only straight P boxes are invertible. See also Substitution box S box Substitution cipher Nothing up my sleeve number Boolean function References references Cryptography navbox block Category Symmetric key cryptography crypto stub fr P Box it P box simple Permutation box ... more details
File Symmetric group 4 permutation list with matrices.svg thumb File Loupe light.svg 15px link 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 Permutations of 4 elements br ... discrete mathematics inversion numbers OEIS A034968 , which have the same parity as the permutation. In mathematics , when X is a finite set of at least two elements, the permutation s of X i.e. ... permutations . If any total ordering of X is fixed, the parity oddness or evenness of a permutation ... mathematics character of the symmetric group S sub n sub . Another notation for the sign of a permutation ... for all maps from X to X , and has value zero for non bijective maps. The sign of a permutation ... of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined . ref name Jacobson Jacobson 2009 , p. 50. ref Example Consider the permutation ... and 4, and finally exchange the places of 1 and 5. This shows that the given permutation is odd. Using the notation explained in the permutation article, we can write math sigma begin pmatrix 1&2&3 .... Properties The identity permutation is an even permutation. ref name Jacobson An even permutation can be obtained from the identity permutation by an even and odd numbers even number of exchanges called transposition mathematics transposition s of two elements, while an odd permutation can ... is even the composition of two odd permutations is even the composition of an odd and an even permutation is odd From these it follows that the inverse of every even permutation is even the inverse of every odd permutation is odd Considering the symmetric group S sub mvar n sub of all permutations of the set ... math here. see https bugzilla.wikimedia.org show bug.cgi?id 1594 c4 that assigns to every permutation ... a given permutation is even or odd, one writes the permutation as a product of disjoint cycles. The permutation ... more details
dablink This article is about a concept in musical set theory . For other uses of this word, see permutation disambiguation . Image P R I RI.png thumb 350px Prime, retrograde, inverse, and retrograde inverse permutations. In music , a permutation of a set music set is any ordering of the elements of that set. ref Allen Forte, The Structure of Atonal Music New Haven and London Yale University Press, 1973 3 John Rahn, Basic Atonal Theory New York Longman, 1980 , 138 ref Different permutations may be related by Transformation music transformation , through the application of zero or more of certain operations , such as transposition music transposition , inversion music inversion , Retrograde music retrogradation , circular permutation also called rotation , or multiplicative operations such as the cycle of fourths and cycle of fifths transforms . These may produce reorderings of the members ... inversions , which are considered a distinct type of permutation. Permutation may be applied to smaller ... result in permutation of the original set. Here is an example of non permutation of trichords, using ... nor Rahn define invariance in this way. Who does? One technique facilitating twelve tone permutation ..., 6, 7, 3, 5, 4, 8, 9, 1, 0 More generally, a musical permutation is any reordering of the prime form ... Atonal Theory New York Longman, 1980 , 137. ref In that regard, a musical permutation is a combinatorics combinatorial permutation from mathematics as it applies to music. Permutations are in no way ... of Paganini for orchestra and piano. Citation needed date March 2011 Cyclical permutation is the maintenance ... be considered a cyclical permutation beginning on the sixth member of a hexachordally combinatorial ... 4, cyclical permutation begins on E music flat 3 in mm. 7 9 Perle 1996, p.20 . See also Counterpoint Identity music Multiplication music Musical set theory Permutation Retrograde music References reflist Twelve tone technique DEFAULTSORT Permutation Music Category Permutations Category Post tonal ... more details
In mathematics, a vexillary permutation is a permutation of the positive integers containing no permutation pattern subpermutation isomorphic to the permutation 2143 in other words, there do not exist four numbers i j k l with j i l k . They were introduced by harvs txt last1 Lascoux last2 Sch tzenberger year1 1982 year2 1985 . The word vexillary means flag like, and comes from the fact that vexillary permutations are related to flag linear algebra flags of module mathematics modules . harvtxt Guibert Pergola Pinzani 2001 showed that vexillary involution mathematics involutions are enumerated by Motzkin number s. References Citation last1 Guibert first1 O. last2 Pergola first2 E. last3 Pinzani first3 R. title Vexillary involutions are enumerated by Motzkin numbers doi 10.1007 PL00001297 mr 1904383 year 2001 journal Annals of Combinatorics issn 0218 0006 volume 5 issue 2 pages 153 174 Citation last1 Lascoux first1 Alain last2 Sch tzenberger first2 Marcel Paul title Polyn mes de Schubert mr 660739 year 1982 journal Comptes Rendus des S ances de l Acad mie des Sciences. S rie I. Math matique issn 0249 6291 volume 294 issue 13 pages 447 450 Citation last1 Lascoux first1 Alain last2 Sch tzenberger first2 Marcel Paul title Schubert polynomials and the Littlewood Richardson rule doi 10.1007 BF00398147 mr 815233 year 1985 journal Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics issn 0377 9017 volume 10 issue 2 pages 111 124 Citation last1 Macdonald first1 I.G. author1 link Ian G. Macdonald title Notes on Schubert polynomials url http books.google.com books?id BvLuAAAAMAAJ publisher Laboratoire de combinatoire et d informatique math matique LACIM , Universit du Qu bec a Montr al series Publications du Laboratoire de combinatoire et d informatique math matique isbn 978 2 89276 086 6 year 1991b volume 6 Category Combinatorics ... more details
In automata theory , a permutation automaton , or pure group automaton , is a deterministic finite automaton such that each input symbol permutation permutes the set of states. ref name McNaughton1967 Citation title The Loop Complexity of Pure Group Events year 1967 month August author McNaughton, Robert journal Information and Control pages 167 176 volume 11 issue 1 2 doi 10.1016 S0019 9958 67 90481 0 ref ref cite journal last Thierrin first Gabriel year 1968 month March title Permutation automata journal Theory of Computing Systems volume 2 issue 1 pages 83 90 doi 10.1007 BF01691347 ref Formally, a deterministic finite automaton mvar A may be defined by the tuple math S , I , &delta , s sub 0 sub , F where mvar S is the set of states of the automaton, mvar I is the set of input symbols, math &delta is the transition function that takes a state mvar s and an input symbol mvar x to a new state math &delta s , x , math s sub 0 sub is the initial state of the automaton, and mvar F is the set of accepting or final states of the automaton. mvar A is a permutation automaton if and only if, for every two distinct states math s sub i sub and math s sub j sub in mvar S and every input symbol mvar x in mvar I , math &delta s sub i sub , x &delta s sub j sub , x . A formal language is p regular also a pure group language if it is accepted by a permutation automaton. For example, the set of strings of even length forms a p regular language it may be accepted by a permutation automaton with two states in which every transition replaces one state by the other. Applications The pure group languages were the first interesting family of regular languages for which the star height problem was proved to be computable . ref name McNaughton1967 ref name Brzozowski80 Janusz Brzozowski computer scientist Janusz A. Brzozowski Open problems about regular languages , In Ronald V. Book, editor, Formal language theory Perspectives and open problems , pp. 23 47. Academic Press, 1980 https ... more details
Programming by permutation , sometimes called programming by accident , is an approach to software development wherein a programming problem is solved by iteratively making small changes permutations and testing each change to see if it behaves as desired. This approach sometimes seems attractive when the programmer does not fully understand the code and believes that one or more small modifications may result in code that is correct. This tactic is rarely productive because a series of small modifications can easily introduce bugs into the code, leading to a solution that is even less correct than the starting point many false starts and corrections usually occur before a satisfactory endpoint is reached it is rarely possible to measure, by empirical testing, whether the solution will work for all cases in the worst case, with poor code management, the original state of the code may be irretrievably lost Programming by permutation gives little or no assurance about the quality of the code produced it is the polar opposite of Formal verification . Programmers are often compelled to program by permutation when an API is insufficiently documented. This lack of clarity drives others to Copy and paste programming copy and paste from reference code which is assumed to be correct, but was itself written as a result of programming by permutation. In some cases where the programmer can logically explain that exactly one out of a small set of variations must work, programming by permutation leads to correct code which then can be verified and makes it unnecessary to think about the other wrong variations. Example For example, the following code sample in C programming language C intended to find and copy a series of digits from a larger string has several problems source lang c char buffer 123abc char destination 10 int i 0 int j 0 int l strlen buffer while i l if isdigit buffer i destination j buffer i i destination j 0 printf s n , destination source First of all, it doesn ... more details
distinguish alternating group In combinatorics combinatorial mathematics , an alternating permutation of the set 1, 2, 3, ..., n is an arrangement of those numbers into an order c sub 1 sub , ..., c sub n sub such that no element c sub i sub is between c sub i &minus 1 sub and c sub i 1 sub for any value of i and c sub 1 sub < c sub 2 sub . In other words, c sub i sub c sub i 1 sub if i is odd and c sub i sub c sub i 1 sub if i is even. For example, the five alternating permutations of 1, 2, 3, 4 are 1, 3, 2, 4 because 1 3 2 4 1, 4, 2, 3 because 1 4 2 3 2, 3, 1, 4 because 2 3 1 4 2, 4, 1, 3 because 2 4 1 3 3, 4, 1, 2 because 3 4 1 2 This type of permutation was first studied by D sir Andr in the 19th century. ref Jessica Millar, N.J.A. Sloane, Neal E. Young, http arxiv.org abs math 0205218v3 A New Operation on Sequences the Boustrouphedon Transform J. Combinatorial Theory, Series A 76 1 44 54 1996 ref If the condition c sub 1 sub < c sub 2 sub is dropped, so we only require that no element c sub i sub is between c sub i &minus 1 sub and c sub i 1 sub , then the permutation is called a zigzag permutation . By exchanging 1 with n , 2 with n &minus 1, etc., each zigzag permutation with c sub 1 sub > c sub 2 sub can be paired uniquely with an alternating permutation. Related integer sequences The determination of the number, A sub n sub , of alternating permutations of the set 1, ..., n is called ... Permutation urlname AlternatingPermutation External links http www.voofie.com content 117 ... more details
1 85798 174 X oclc 30834713 preceded by followed by Permutation City is a 1994 science fiction novel ... INTERVIEWS Interviews.html Aurealis ref Permutation City won the Campbell award ... Universes ref Themes and setting Original research date October 2008 Permutation City asks whether there is a difference ... s Ladder deal with related issues from various other perspectives. Story The plot of Permutation ... VR city, Permutation City, the largest VR environment ever conceived, complete with reactive crowds ... hack them both, along with any moderately sized software packages they wish, into Permutation City ... its true identity to be fulfilled. Part two Maria wakes in Permutation City seven thousand years ... hydrophilus , and the citizens of Permutation City were on the verge of making contact with the creatures ... had independently hypothesized the existence of a creator. Meanwhile, Permutation City had ... growing chunks of the TVC universe, with Permutation City acting as the central locus for interaction ... of Permutation City, were now progressively being redefined in ways which prevented them from interfering ... Maria had initially designed. To the alarm of its citizens, Permutation City and eventually the entire ... its possibility. As Permutation City becomes corrupted and the TVC universe begins to suffer ... rules that governed the Autoverse s takeover of Permutation City. See also portal Novels Mind uploading Simulated reality References references External links http www.gregegan.net PERMUTATION Permutation.html ... Permutaci n fr La Cit des permutants it Permutation City ... more details
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