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Superpermutation
In combinatorial mathematics, a superpermutation on ''n'' symbols is a string that contains each permutation of ''n'' symbols as a substring. While trivial superpermutations can simply be made up of every permutation listed together, superpermutations can also be shorter (except for the trivial case of ''n'' = 1) because overlap is allowed. For instance, in the case of ''n'' = 2, the superpermutation 1221 contains all possible permutations (12 and 21), but the shorter string 121 also contains both permutations. It has been shown that for 1 ≤ ''n'' ≤ 5, the smallest superpermutation on ''n'' symbols has length 1! + 2! + … + ''n''! . The first four smallest superpermutations have respective lengths 1, 3, 9, and 33, forming the strings 1, 121, 123121321, and 123412314231243121342132413214321. However, for ''n'' = 5, there are several smallest superpermutations having the length 153. One such superpermutation is shown below, while another of the same length can be obtained by s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superpermutation Distribution
In combinatorial mathematics, a superpermutation on ''n'' symbols is a string that contains each permutation of ''n'' symbols as a substring. While trivial superpermutations can simply be made up of every permutation listed together, superpermutations can also be shorter (except for the trivial case of ''n'' = 1) because overlap is allowed. For instance, in the case of ''n'' = 2, the superpermutation 1221 contains all possible permutations (12 and 21), but the shorter string 121 also contains both permutations. It has been shown that for 1 ≤ ''n'' ≤ 5, the smallest superpermutation on ''n'' symbols has length 1! + 2! + … + ''n''! . The first four smallest superpermutations have respective lengths 1, 3, 9, and 33, forming the strings 1, 121, 123121321, and 123412314231243121342132413214321. However, for ''n'' = 5, there are several smallest superpermutations having the length 153. One such superpermutation is shown below, while another of the same length can be obtained by s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superpermutations
In combinatorial mathematics, a superpermutation on ''n'' symbols is a string that contains each permutation of ''n'' symbols as a substring. While trivial superpermutations can simply be made up of every permutation listed together, superpermutations can also be shorter (except for the trivial case of ''n'' = 1) because overlap is allowed. For instance, in the case of ''n'' = 2, the superpermutation 1221 contains all possible permutations (12 and 21), but the shorter string 121 also contains both permutations. It has been shown that for 1 ≤ ''n'' ≤ 5, the smallest superpermutation on ''n'' symbols has length 1! + 2! + … + ''n''! . The first four smallest superpermutations have respective lengths 1, 3, 9, and 33, forming the strings 1, 121, 123121321, and 123412314231243121342132413214321. However, for ''n'' = 5, there are several smallest superpermutations having the length 153. One such superpermutation is shown below, while another of the same length can be obtained by s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greg Egan
Greg Egan (born 20 August 1961) is an Australian science fiction writer and amateur mathematician, best known for his works of hard science fiction. Egan has won multiple awards including the John W. Campbell Memorial Award, the Hugo Award, and the Locus Award. Life and work Egan holds a Bachelor of Science degree in Mathematics from the University of Western Australia. He published his first work in 1983. He specialises in hard science fiction stories with mathematical and quantum ontology themes, including the nature of consciousness. Other themes include genetics, simulated reality, posthumanism, mind uploading, sexuality, artificial intelligence, and the superiority of rational naturalism to religion. He often deals with complex technical material, like new physics and epistemology. He is a Hugo Award winner (with eight other works shortlisted for the Hugos) and has also won the John W. Campbell Memorial Award for Best Science Fiction Novel. His early stories feature s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Bruijn Sequence
In combinatorial mathematics, a de Bruijn sequence of order ''n'' on a size-''k'' alphabet ''A'' is a cyclic sequence in which every possible length-''n'' string on ''A'' occurs exactly once as a substring (i.e., as a ''contiguous'' subsequence). Such a sequence is denoted by and has length , which is also the number of distinct strings of length ''n'' on ''A''. Each of these distinct strings, when taken as a substring of , must start at a different position, because substrings starting at the same position are not distinct. Therefore, must have ''at least'' symbols. And since has ''exactly'' symbols, De Bruijn sequences are optimally short with respect to the property of containing every string of length ''n'' at least once. The number of distinct de Bruijn sequences is :\dfrac. The sequences are named after the Dutch mathematician Nicolaas Govert de Bruijn, who wrote about them in 1946. As he later wrote, the existence of de Bruijn sequences for each order together ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superpattern
In the mathematical study of permutations and permutation patterns, a superpattern or universal permutation is a permutation that contains all of the patterns of a given length. More specifically, a ''k''-superpattern contains all possible patterns of length ''k''. Definitions and example If π is a permutation of length ''n'', represented as a sequence of the numbers from 1 to ''n'' in some order, and ''s'' = ''s''1, ''s''2, ..., ''s''''k'' is a subsequence of π of length ''k'', then ''s'' corresponds to a unique ''pattern'', a permutation of length ''k'' whose elements are in the same order as ''s''. That is, for each pair ''i'' and ''j'' of indexes, the ''i''th element of the pattern for ''s'' should be less than the ''j''the element if and only if the ''i''th element of ''s'' is less than the ''j''th element. Equivalently, the pattern is order-isomorphic to the subsequence. For instance, if π is the permutation 25314, then it has ten subsequences of length three, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haruhi Suzumiya
is a Japanese light novel series written by Nagaru Tanigawa and illustrated by Noizi Ito. It was first published in 2003 by Kadokawa Shoten in Japan with the novel ''The Melancholy of Haruhi Suzumiya'', and has since been followed by 11 additional novel volumes, an anime television series adaptation produced by Kyoto Animation, four manga series, an animated film, two original net animation series and several video games. After the anime adaptation airing in 2006, publishing company Kadokawa Shoten received offers for licensing the novels and their adaptations. The novels are licensed for English language release in the United States by Little, Brown and Company, for young readers by Yen Press and the anime adaptation was licensed for North American distribution by Kadokawa Pictures USA division which then sub-licensed production and distribution to Bandai Entertainment. The anime is currently licensed by Crunchyroll. Plot Kyon is a student at North High Schoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics On Words
Combinatorics on words is a fairly new field of mathematics, branching from combinatorics, which focuses on the study of words and formal languages. The subject looks at letters or symbols, and the sequences they form. Combinatorics on words affects various areas of mathematical study, including algebra and computer science. There have been a wide range of contributions to the field. Some of the first work was on square-free words by Axel Thue in the early 1900s. He and colleagues observed patterns within words and tried to explain them. As time went on, combinatorics on words became useful in the study of algorithms and coding. It led to developments in abstract algebra and answering open questions. Definition Combinatorics is an area of discrete mathematics. Discrete mathematics is the study of countable structures. These objects have a definite beginning and end. The study of enumerable objects is the opposite of disciplines such as analysis, where calculus and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brady Haran
Brady John Haran (born 18 June 1976) is an Australian-British independent filmmaker and video journalist who produces educational videos and documentary films for his YouTube channels, the most notable being ''Periodic Videos'' and ''Numberphile''. Haran is also the co-host of the'' Hello Internet'' podcast along with fellow educational YouTuber CGP Grey. On 22 August 2017, Haran launched his second podcast, called ''The Unmade Podcast'', and on 11 November 2018, he launched his third podcast, '' The Numberphile Podcast'', based on his mathematics-centered channel of the same name. Reporter and filmmaker Brady Haran studied journalism for a year before being hired by ''The Adelaide Advertiser''. In 2002, he moved from Australia to Nottingham, United Kingdom. In Nottingham, he worked for the BBC, began to work with film, and reported for ''East Midlands Today'', BBC News Online and BBC radio stations. In 2007, Haran worked as a filmmaker-in-residence for Nottingham Science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Pattern
In combinatorial mathematics and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation as a sequence of digits representing the result of applying the permutation to the digit sequence 123...; for instance the digit sequence 213 represents the permutation on three elements that swaps elements 1 and 2. If π and σ are two permutations represented in this way (these variable names are standard for permutations and are unrelated to the number pi), then π is said to ''contain'' σ as a ''pattern'' if some subsequence of the digits of π has the same relative order as all of the digits of σ. For instance, permutation π contains the pattern 213 whenever π has three digits ''x'', ''y'', and ''z'' that appear within π in the order ''x''...''y''...''z'' but whose values are ordered as ''y'' < ''x'' < ''z'', the same as the ordering of the values in the permutation 213. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs. Definition Let G be a group and S be a generating set of G. The Cayley graph \Gamma = \Gamma(G,S) is an edge-colored directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405. * Each element g of G is assigned a vertex: the vertex set of \Gamma is identified with G. * Each element s of S is assigned a color c_s. * For every g \in G and s \in S, there is a directed edge of color c_s from the vertex corresponding to g to the one corresponding to gs. Not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
