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Pattern Avoidance
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] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
European Journal Of Combinatorics
European, or Europeans, or Europeneans, may refer to: In general * ''European'', an adjective referring to something of, from, or related to Europe ** Ethnic groups in Europe ** Demographics of Europe ** European cuisine European cuisine comprises the cuisines of Europe "European Cuisine." [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanley–Wilf Conjecture
The Stanley–Wilf conjecture, formulated independently by Richard P. Stanley and Herbert Wilf in the late 1980s, states that the growth rate of every proper permutation class is singly exponential. It was proved by and is no longer a conjecture. Marcus and Tardos actually proved a different conjecture, due to , which had been shown to imply the Stanley–Wilf conjecture by . Statement The Stanley–Wilf conjecture states that for every permutation ''β'', there is a constant ''C'' such that the number , ''S''''n''(''β''), of permutations of length ''n'' which avoid ''β'' as a permutation pattern is at most ''C''''n''. As observed, this is equivalent to the convergence of the limit :\lim_ \sqrt The upper bound given by Marcus and Tardos for ''C'' is exponential in the length of ''β''. A stronger conjecture of had stated that one could take ''C'' to be , where ''k'' denotes the length of ''β'', but this conjecture was disproved for the permutation by . Indeed, has sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Wilf
Herbert Saul Wilf (June 13, 1931 – January 7, 2012) was a mathematician, specializing in combinatorics and graph theory. He was the Thomas A. Scott Professor of Mathematics in Combinatorial Analysis and Computing at the University of Pennsylvania. He wrote numerous books and research papers. Together with Neil Calkin he founded ''The Electronic Journal of Combinatorics'' in 1994 and was its editor-in-chief until 2001. Biography Wilf was the author of numerous papers and books, and was adviser and mentor to many students and colleagues. His collaborators include Doron Zeilberger and Donald Knuth. One of Wilf's former students is Richard Garfield, the creator of the collectible card game ''Magic: The Gathering''. He also served as a thesis advisor for E. Roy Weintraub in the late 1960s. Wilf died of a progressive neuromuscular disease in 2012. Awards In 1998, Wilf and Zeilberger received the Leroy P. Steele Prize for Seminal Contribution to Research for their joint pap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard P
Richard is a male given name. It originates, via Old French, from Frankish language, Old Frankish and is a Compound (linguistics), compound of the words descending from Proto-Germanic language, Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick (nickname), Dick", "Dickon", "Dickie (name), Dickie", "Rich (given name), Rich", "Rick (given name), Rick", "Rico (name), Rico", "Ricky (given name), Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, [...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 punched cards ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Applied Mathematics
''Advances in Applied Mathematics'' is a peer-reviewed mathematics journal publishing research on applied mathematics. Its founding editor was Gian-Carlo Rota (Massachusetts Institute of Technology); from 1980 to 1999, Joseph P. S. Kung (University of North Texas) served as managing editor. It is currently published by Elsevier with eight issues per year and edited by Hal Schenck (Auburn University) and Catherine Yan (Texas A&M University). Abstracting and indexing The journal is abstracted and indexed by: * ACM Guide to Computing Literature * CompuMath Citation Index * Current Contents/Physics, Chemical, & Earth Sciences * ''Mathematical Reviews'' * Science Citation Index * Scopus According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.848. See also * List of periodicals published by Elsevier This is a list of scientific, technical and general interest periodicals published by Elsevier or one of its imprints or subsidiary companies. Both pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Algebraic Combinatorics
''Journal of Algebraic Combinatorics'' is a peer-reviewed scientific journal covering algebraic combinatorics Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algeb .... It was established in 1992 and is published by Springer Science+Business Media. The editor-in-chief is Ilias S. Kotsireas ( Wilfrid Laurier University). In 2017, the journal's four editors-in-chief and editorial board resigned to protest the publisher's high prices and limited accessibility. They criticized Springer for "double-dipping", that is, charging large subscription fees to libraries in addition to high fees for authors who wished to make their publications open access. The board subsequently started their own open access journal, ''Algebraic Combinatorics''. Abstracting and indexing The journal is abstracted a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum Of Permutations
In combinatorics, the skew sum and direct sum of permutations are two operations to combine shorter permutations into longer ones. Given a permutation ''π'' of length ''m'' and the permutation ''σ'' of length ''n'', the skew sum of ''π'' and ''σ'' is the permutation of length ''m'' + ''n'' defined by : (\pi\ominus\sigma)(i) = \begin \pi(i)+n & \text1\le i\le m, \\ \sigma(i-m) & \textm+1\le i\le m+n,\end and the direct sum of ''π'' and ''σ'' is the permutation of length ''m'' + ''n'' defined by : (\pi\oplus\sigma)(i) = \begin \pi(i) & \text1\le i\le m,\\ \sigma(i-m)+m & \textm+1\le i\le m+n.\end Examples The skew sum of the permutations ''π'' = 2413 and ''σ'' = 35142 is 796835142 (the last five entries are equal to ''σ'', while the first four entries come from shifting the entries of ''π'') while their direct sum is 241379586 (the first four entries are equal to ''π'', while the last five come from shifting the entries of ''σ''). Sums of perm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The very first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.87. Notable publications * The 1972 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Matrices
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when used to multiply another matrix, say , results in permuting the rows (when pre-multiplying, to form ) or columns (when post-multiplying, to form ) of the matrix . Definition Given a permutation of ''m'' elements, :\pi : \lbrace 1, \ldots, m \rbrace \to \lbrace 1, \ldots, m \rbrace represented in two-line form by :\begin 1 & 2 & \cdots & m \\ \pi(1) & \pi(2) & \cdots & \pi(m) \end, there are two natural ways to associate the permutation with a permutation matrix; namely, starting with the ''m'' × ''m'' identity matrix, , either permute the columns or permute the rows, according to . Both methods of defining permutation matrices appear in the literature and the properties expressed in one representation can be easily converted to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
