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Superpermutation
In combinatorics, combinatorial mathematics, a superpermutation on ''n'' symbols is a string (computer science), string that contains each permutation of ''n'' symbols as a substring. While Triviality (mathematics), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superpermutation Distribution
In combinatorics, combinatorial mathematics, a superpermutation on ''n'' symbols is a string (computer science), string that contains each permutation of ''n'' symbols as a substring. While Triviality (mathematics), 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 ... [...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 ... [...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] |
4chan
4chan is an anonymous English-language imageboard website. Launched by Christopher "moot" Poole in October 2003, the site hosts boards dedicated to a wide variety of topics, from anime and manga to video games, cooking, weapons, television, music, literature, history, fitness, politics, and sports, among others. Registration is not available and users typically post anonymously. , 4chan receives more than 22 million unique monthly visitors, of which approximately half are from the United States. 4chan was created as an unofficial English-language counterpart to the Japanese imageboard Futaba Channel, also known as 2chan, and its first boards were created for posting images and discussion related to anime. The site has been described as a hub of Internet subculture, its community being influential in the formation and popularization of prominent Internet memes, such as lolcats, Rickrolling, rage comics, wojaks, Pepe the Frog, as well as hacktivist and political move ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substring
In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, "''the best of''" is a substring of "''It was the best of times''". In contrast, "''Itwastimes''" is a subsequence of "''It was the best of times''", but not a substring. Prefixes and suffixes are special cases of substrings. A prefix of a string S is a substring of S that occurs at the beginning of S; likewise, a suffix of a string S is a substring that occurs at the end of S. The substrings of the string "''apple''" would be: "''a''", "''ap''", "''app''", "''appl''", "''apple''", "''p''", "''pp''", "''ppl''", "''pple''", "''pl''", "''ple''", "''l''", "''le''" "''e''", "" (note the empty string at the end). Substring A string u is a substring (or factor) of a string t if there exists two strings p and s such that t = pus. In particular, the empty string is a substring of every string. Example: The string u=ana is equal to substrings (and subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On-Line Encyclopedia Of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is chairman of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 350,000 sequences, making it the largest database of its kind. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. History Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics. The database was at first stored on punche ... [...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 inf ... [...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] |
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] |
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. ... [...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] |
