A cyclicpermutation or circular permutation is a permutation built from one or more Set mathematics sets of elements in cyclic order . The notion cyclicpermutation 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 cyclicpermutation 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 may be written as p c i c i t for i 1, 2, ..., k &minus t , and p c i c i t &minus k for i k &minus t 1, k &minus t 2, ..., k . Note Every cyclicpermutation of definition ... & 1 & 2 end pmatrix 1356 2478 math is a cyclicpermutation with offset 2. It may be constructed with gcd ... perm 2.png right mapping of permutation A permutation is called a cyclicpermutation if and only if it will be constructed with exactly 1 cycle. Note Every permutation over a set with k elements is a cyclicpermutation of definition type 2 if and only if it is a cyclicpermutation of definition ... & 7 & 1 end pmatrix 14625837 math Definition 3 image 050712 perm 3.png right mapping of permutation A permutation is called a cyclicpermutation if and only if only one of the constructing cycles will have length 1. Note Every cyclicpermutation of definition type 3 may be seen as an union mathematics union of a cyclicpermutation of definition type 2 and some fixed point mathematics fixed points . Every cyclicpermutation of definition type 2 may be seen as a cyclicpermutation of definition type ... end pmatrix 146837 2 5 math See also Cyclicpermutation of integer Cycle notation Cycles and fixed ... of equal length see cycles and fixed points . Cyclic permutations of definition type 1 are also called ... fr Permutation circulaire mk pt Permuta o circular sr sv Cyklisk permutation ... more details
Convolution Cyclic order Cyclicpermutation Even and odd permutations Factorial number system ...Other uses Lead rewrite date August 2010 In mathematics , the notion of permutation is used with several ... objects or values. Informally, a permutation of a set of objects is an arrangement of those objects ... 1,2,3 , 1,3,2 , 2,1,3 , 2,3,1 , 3,1,2 , and 3,2,1 nowiki . One might define an anagram of a word as a permutation ... of sorting algorithm s in computer science. In algebra and particularly in group theory , a permutation ... certain permutations related to it. Generalities The notion of permutation is used in the following ... 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 ... In combinatorics , a permutation is usually understood to be a sequence containing each element ... have no order 1, 2, 3 and 3, 2, 1 are different ways to denote the same set. In this sense a permutation ... n permutations of S . There is also a weaker meaning of the term permutation that is sometimes used ..., more common, meanings of permutation . The number of such k permutations of n is denoted variously ... n sup u k u sup of n . If M is a finite multiset , then a multiset permutation is a sequence of elements ... Permutations in group theory Main Symmetric group In group theory, the term permutation of a set means ... of degree n . Any subgroup of a symmetric group is called a permutation group . In fact by Cayley s theorem any group is isomorphic to a permutation group, and every finite group to a subgroup of a finite symmetric group. However, permutation groups have more structure than abstract groups, and different realizations of a group as a permutation group therefore need not be equivalent. Notation ... 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 math ... 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 ... written in Cycle notation cyclic form ref e.g. during cycle index computations ref so that given the set 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 permutation group by virtue of its regular group ... 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 sup &minus ... 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 , 13 24 , 14 ... 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
confusing date June 2010 Unreferenced date 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 math stub ... more details
In cryptography , the term pseudorandom permutation , abbreviated PRP , refers to a function that cannot be distinguished from a random permutation 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 random permutation. 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 Permutation box 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 crypto stub Category Theory of cryptography ... more details
Context date October 2009 A permutation automaton or p automaton is an automata theory automaton such that each input permutation permutes the set of states. In other words, a permutation automaton is a reset free deterministic finite automaton . Formally, an automaton math A S, I, delta, s 0, F math is a permutation automaton if and only if 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 url http www.springerlink.com content p2746l6r4468j251 accessdate 2009 01 23 doi 10.1007 BF01691347 ref math forall s i, s j in S forall a in I delta s i,a delta s j,a implies s i s j math A formal language language is p regular if it is accepted by a permutation automaton. References references Category Automata theory Category Formal languages Category Permutations formalmethods 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 ... of permutation graphs can be computed in polynomial time these algorithms exploit the fact that the number of Vertex separator Minimal separators inclusion minimal vertex separators in a permutation ... ref Relation to other graph classes Permutation graphs are a special case of circle graph s, comparability ... more details
A random permutation 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 that use randomized algorithm s. Such fields include coding theory , cryptography , and simulation . A good example of a random permutation is the shuffling of a card deck deck of cards this is ideally a random permutation of the 52 cards. One method of generating a random permutation 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 & n x 1 & x 2 & x 3 & cdots & x n end pmatrix . math The above brute force method will require occasional retries whenever the random number picked is a repeat of a number already selected. A simple algorithm to generate a permutation of n items uniformly at random without retries, known as the Knuth shuffle , is to start with the Identity function identity permutation or any other permutation, and then go through the positions 1 through n 1, and for each position i swap the element currently there with an arbitrarily chosen element from positions i through n , inclusive. It s easy to verify that any permutation of n elements will be produced by this algorithm with probability ... distributed random permutation, see rencontres numbers . That distribution approaches a Poisson ... Dickman constant Perfect shuffle Random permutation statistics Shuffle Shuffling algorithms Shuffling ... RandomPermutation.html Random permutation at MathWorld http www.techuser.net randpermgen.html Random permutation generation detailed and practical explanation of Knuth shuffle algorithm and its variants ... Category Randomness fr Permutation al atoire fi Satunnainen permutaatio ... 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 ... 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, 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 ... thus the symmetric group is not abelian. br These are permutation matrices as well. small In mathematics , in matrix theory , a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0 s 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 & 2 & cdots & m pi 1 & pi 2 & cdots & pi m end pmatrix , math its permutation matrix is the m × ... permutation matrices P sub &pi sub and P sub &sigma sub math P sigma P pi P pi circ sigma math ... pi i math as rows of the permutation matrix if one had used columns instead then the product above would have been equal to math P sigma circ pi math with the permutations in their original order. As permutation ..., on 1,2,..., n . Since there are n permutations, there are n permutation matrices. By the formulas above, the n × n permutation matrices form a Group mathematics group under matrix multiplication with the identity matrix as the identity 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 matrix is a doubly stochastic matrix . The Birkhoff von Neumann theorem says that every doubly stochastic matrix is a convex combination of permutation matrices of the same order and the permutation matrices are the extreme point s of the set of doubly stochastic ... hull of the set of permutation matrices. The product PM , premultiplying a matrix M by a permutation ... . Thus, A n . The Trace linear algebra trace of a permutation matrix is the number of fixed points ... 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 random permutation 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 bigrams and trigrams 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 anagramming . See also Topics in cryptography Crypto navbox classical Category Classical ciphers Category Permutations hu Permut ci s ... more details
In mathematics , when X is a finite set of at least two elements, the permutation s of X i.e. the bijective ... and the odd permutations . If any total ordering of X is fixed, the parity oddness or evenness of a permutation ... x y math and math sigma x sigma y math . The sign or signature of a permutation math sigma math ... n sub . Another notation for the sign of a permutation is given by the more general Levi Civita ... bijective maps. The sign of a permutation can be explicitly expressed as math sgn sigma 1 N sigma ... of a permutation math sigma math can be defined from its decomposition into the product of transposition ... of transpositions in all decompositions is the same, implying that the sign of a permutation is well defined . ref Jacobson 2009 , p. 50. ref Example Consider the permutation &sigma of the set 1,2,3,4,5 .... This shows that the given permutation &sigma is odd. Using the notation explained in the permutation ... permutation is an even permutation. ref Jacobson 2009 , p. 50. ref 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 be obtained by an odd ... 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 n sub of all permutations of the set ... permutation its signature is a group homomorphism . Furthermore, we see that the even permutations ... permutation is even or odd, one writes the permutation as a product of disjoint cycles. The permutation ... for determining whether a given permutation is even or odd is to construct the corresponding Permutation matrix and compute its determinant. The value of the determinant is same as the parity of the permutation. Every permutation of odd order group theory order must be even the converse is not true ... more details
dablink This article is about a concept in musical set theory . For other uses of this word, see permutation disambiguation . In music , a permutation of a set music set is a Transformation music transformation of its prime form by applying zero or more of certain operations , specifically transposition music transposition , inversion music inversion , and retrograde . This produces a reordering of the members of the set. The permutations resulting from applying the inversion or retrograde operations are categorized as the prime form s inversions and retrogrades , respectively. Likewise, applying both inversion and retrograde to a prime form produces its retrograde inversions , which are considered a distinct type of permutation. Here is an example of permutation usage in the tone row or twelve tone series from Anton Webern s Concerto Webern Concerto Image Webern Concerto Op. 24 tone row.png thumb center 550px Webern s Concerto Op. 24 tone row, ref Whittall, Arnold. 2008. The Cambridge Introduction to Serialism. Cambridge Introductions to Music , p.97. New York Cambridge University Press ... music invariance . One technique facilitating twelve tone permutation is the use of number ..., 0 More generally, a musical permutation is any reordering of the prime form of an ordered set of pitch ... 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 limited to the twelve ... and piano. Citation needed date March 2011 Cyclical permutation is the maintenance of the original ... permutation beginning on the sixth member of a hexachordally combinatorial row. The tone row from ... statement.png thumb center 350px Initial statement begins on F 5 , mm. 2 4, cyclical permutation ... music Musical set theory Permutation References reflist Twelve tone technique DEFAULTSORT Permutation Music Category Permutations Category Post tonal music theory Category Musical set theory ... 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 citation title Permutation Polynomial Interleavers ... Codes using Permutation Polynomials over Integer Rings year 2005 first1 Oscar last1 Takeshita id arxiv archive cs id 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 ... sees math g 0 0 g 1 3 g 2 2 g 3 1 math , so the polynomial defines the permutation math begin pmatrix ... , so the polynomial defines the permutation math begin pmatrix 0 &1 & 2 & 3 & 4 & 5 & 6 & 7 0 &3 ... Z p sup k sup Z . Lemma for k 1 i.e. Z p Z such polynomial defines a permutation only in the case ... Z p sup k sup Z such polynomial defines a permutation if and only if math a 0 mod p math and math b ... numbers. Lemma any polynomial math g x a 0 sum 0 i le M a i x i math defines a permutation for the ring ... permutation polynomials using the following simple construction. Consider math n p 1 k 1 p 2 k 2 ...p ... and math b 1 math . Then such polynomial defines a permutation. To see this we observe that for all ... linear polynomial 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 ... 0 mod p math i 0 , then polynomial g x defines a permutation for the elements of the ring Z p sup ... prime number p . The cubic polynomial math g x ax 3 bx math defines a permutation if and only if for all ... reciprocity law . So one can see that the analysis of higher degree polynomials to define a permutation ... 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 Crypto navbox block Category Cryptography crypto stub it P box simple Permutation box ... 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
Programming by permutation 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 expected. 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 t give the right answer. With the given starting ... 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 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 Andr s problem .... Further reading MathWorld title Alternating Permutation urlname AlternatingPermutation External links ... 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 is there a difference .... Story The plot of Permutation City follows the lives of several people in a near future reality ... Carter, to build a full scale, high resolution VR city, Permutation City, the largest VR environment ... packages they wish, into Permutation City s machine code , guaranteeing them a place in the city .... Part Two Maria wakes in Permutation City seven thousand years of subjective time after the launch ... of Permutation City were on the verge of making contact with the creatures. However, a town hall ... hypothesized the existence of a creator. Meanwhile, Permutation City had flourished in the TVC universe ... universe, with Permutation City acting as the central locus for interaction between populations located ... had previously done the work of simulating the Autoverse for the inhabitants of Permutation City, were ... designed. To the alarm of its citizens, Permutation City and eventually the entire TVC processor .... As Permutation City becomes corrupted and the TVC universe begins to suffer spontaneous systemic failures ... s takeover of Permutation City. See also portal Novels Mind uploading Simulated reality References references External links http www.gregegan.net PERMUTATION Permutation.html Official site ... culture cs M sto permutac es Ciudad Permutaci n fr La Cit des permutants it Permutation City ... more details
There are many terms in mathematics that begin with cyclicCyclic chain rule , for derivatives, used in thermodynamics Cyclic code , linear codes closed under cyclic permutations Cyclic convolution , a method of combining periodic functions Cycle decomposition disambiguation Cyclic extension , a field extension with cyclic Galois group Cycle graph or cyclic graph is a connected, 2 regular graph Cycle graph algebra , a diagram representing the cycles determined by taking powers of group elements Circulant graph , a graph whose adjacency matrix is circulant Cycle graph theory , a nontrivial path from a node to itself Cyclic group , a group generated by a single element Cyclic homology , an approximation of K theory used in non commutative differential geometry Cyclic module , a module generated by a single element Cyclic notation , a way of writing permutations Cyclic number , a number such that cyclic permutations of the digits are successive multiples of the number Cyclic order , a binary relation for doubly linked lists Cyclicpermutation , a permutation with one nontrivial orbit Cyclic polygon , a polygon which can be given a circumscribed circle Cyclic shift , also known as circular shift Cyclic symmetry , n fold rotational symmetry of 3 dimensional space See also Cycle disambiguation Cycle mathematics Category Mathematics related lists sv Cyklisk matematik ... more details
concept cyclic number group theory summary in Repeating decimal Mergefrom Transposable integer discuss Talk Cyclic number Cyclicpermutation of integer date September 2009 A cyclic number is an integer in which cyclicpermutation s of the digits are successive multiples of the number. The most widely ... Cyclic numbers are related to the Repeating decimal recurring digital representations of unit fractions . A cyclic number of length L is the digital representation of 1 L 1 . Conversely, if the digital period of 1 p where p is prime is p &minus 1, then the digits represent a cyclic number. For example 1 7 0.142857 142857 . Multiples of these fractions exhibit cyclicpermutation 1 7 0.142857 ... in hexadecimal . See also Repeating decimal Fermat s little theorem Cyclicpermutation of integer ... 142857 × 4 571428 142857 × 5 714285 142857 × 6 857142 Details To qualify as a cyclic number, it is required that successive multiples be cyclic permutations. Thus, the number 076923 would not be considered a cyclic number, even though all cyclic permutations are multiples 076923 ... 5 repeated digits, i.e. 555 repeated cyclic numbers, i.e. 142857142857 If leading zeros are not permitted on numerals, then 142857 is the only cyclic number in decimal . Allowing leading zeros, the sequence of cyclic numbers begins 142857 6 digits 0588235294117647 16 digits 052631578947368421 18 ... 857142 . Form of cyclic numbers From the relation to unit fractions, it can be shown that cyclic numbers ... is a Prime number prime that does not Divisor divide b . Primes p that give cyclic numbers are called full reptend prime s or long primes . For example, the case b 10, p 7 gives the cyclic number 142857. Not all values of p will yield a cyclic number using this formula for example p 13 gives 076923076923 ... values of p for which this formula produces cyclic numbers in decimal are sequence OEIS A001913 A001913 .... Construction of cyclic numbers Cyclic numbers can be constructed by the following Algorithm ... more details
Image Cyclic adenosine monophosphate 2D skeletal.png thumb Cyclic adenosine monophosphate Image CGMP.png thumb Cyclic guanosine monophosphate A cyclic nucleotide is any nucleotide in which the phosphate group is bonded to two of the sugar s hydroxyl groups, forming a cyclical or ring structure. These include cyclic AMP cyclic GMP cyclic ADP ribose These function as second messenger s associated with G protein s and calcium signaling . External links MeshName Nucleotides, Cyclic Nucleobases, nucleosides, and nucleotides Category Nucleotides Biochem stub et Ts klilised nukleotiidid nl Cyclisch nucleotide no Syklisk nukleotid sr Cikli ni nukleotid ... more details
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