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Hook Length Formula
In combinatorial mathematics, the hook length formula is a formula for the number of standard Young tableaux whose shape is a given Young diagram. It has applications in diverse areas such as representation theory, probability, and algorithm analysis; for example, the problem of longest increasing subsequences. A related formula gives the number of semi-standard Young tableaux, which is a specialization of a Schur polynomial. Definitions and statement Let \lambda=(\lambda_1\geq \cdots\geq \lambda_k) be a partition of n=\lambda_1+\cdots+\lambda_k. It is customary to interpret \lambda graphically as a Young diagram, namely a left-justified array of square cells with k rows of lengths \lambda_1,\ldots,\lambda_k. A (standard) Young tableau of shape \lambda is a filling of the n cells of the Young diagram with all the integers \, with no repetition, such that each row and each column form increasing sequences. For the cell in position (i,j), in the ith row and jth column, the hook H_ ... [...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] |
Gilbert De Beauregard Robinson
Gilbert de Beauregard Robinson, MBE (3 June 1906 – 8 April 1992) was a Canadian mathematician most famous for his work on combinatorics and representation theory of the symmetric groups, including the Robinson-Schensted algorithm. Biography Gilbert Robinson was born in Toronto in 1906. He then attended St. Andrew's College and graduated from the University of Toronto in 1927. He received his Ph.D at Cambridge where his advisor was group theorist Alfred Young. He then joined the Mathematics Department in Toronto where he served until his retirement in 1971, except for a period of wartime service in Ottawa. Robinson specialized in the study of the symmetric groups on which he became a recognized authority. In 1938 he formulated, in a paper studying the Littlewood–Richardson rule, a correspondence that would later become known as the Robinson-Schensted correspondence. He wrote some forty papers on the topic of symmetric groups. He also published ''The Foundations of Geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kostka Number
In mathematics, the Kostka number ''K''λμ (depending on two Partition (number theory), integer partitions λ and μ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ and weight μ. They were introduced by the mathematician Carl Kostka in his study of symmetric functions (). For example, if λ = (3, 2) and μ = (1, 1, 2, 1), the Kostka number ''K''λμ counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows. The three such tableaux are shown at right, and ''K''(3, 2) (1, 1, 2, 1) = 3. Examples and special cases For any partition λ, the Kostka number ''K''λλ is equal to 1: the unique way to fill the Young diagram of shape λ = (λ1, λ2, ..., λ''m'') with λ1 copies of 1, λ2 copies of 2, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Specht Module
In mathematics, a Specht module is one of the representations of symmetric groups studied by . They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of ''n'' form a complete set of irreducible representations of the symmetric group on ''n'' points. Definition Fix a partition λ of ''n'' and a commutative ring ''k''. The partition determines a Young diagram with ''n'' boxes. A Young tableau of shape λ is a way of labelling the boxes of this Young diagram by distinct numbers 1, \dots, n. A tabloid is an equivalence class of Young tableaux where two labellings are equivalent if one is obtained from the other by permuting the entries of each row. For each Young tableau ''T'' of shape λ let \ be the corresponding tabloid. The symmetric group on ''n'' points acts on the set of Young tableaux of shape λ. Consequently, it acts on tabloids, and on the free ''k''-module ''V'' with the tabloids as basis. Given a Young table ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of The Symmetric Group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids. The symmetric group S''n'' has order ''n''!. Its conjugacy classes are labeled by partitions of ''n''. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of ''n''. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representations by the same set that parametrizes conjugacy classes, namely by partitions of ''n'' or equivalently Young diagrams of size ''n''. Each such irreducible representation can in fact be realized over the integers (every permutation acting by a mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corners Of Young
{{disambiguation ...
Corners may refer to: * A community formed at a crossroads or other intersection; a few examples include: ** Balcom Corners, New York ** Bells Corners in Ottawa ** Dixon's Corners, Ontario ** Five Corners, Wisconsin (other), any of three communities of that name ** Hales Corners, Wisconsin ** Hallers Corners, Michigan ** Layton Corners, Michigan * Corners, a variation on the Four Seasons card game * ''Corners'' (TV series), 1980s BBC children's television series * Corners, Perry County, Missouri, an unincorporated community See also * Corner (other) Corner may refer to: People *Corner (surname) *House of Cornaro, a noble Venetian family (''Corner'' in Venetian dialect) Places *Corner, Alabama, a community in the United States *Corner Inlet, Victoria, Australia *Corner River, a tributary of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer science. Knuth has been called the "father of the analysis of algorithms". He is the author of the multi-volume work ''The Art of Computer Programming'' and contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process, he also popularized the asymptotic notation. In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar, Knuth created the WEB and CWEB computer programming systems designed to encou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igor Pak
Igor Pak (russian: link=no, Игорь Пак) (born 1971, Moscow, Soviet Union) is a professor of mathematics at the University of California, Los Angeles, working in combinatorics and discrete probability. He formerly taught at the Massachusetts Institute of Technology and the University of Minnesota, and he is best known for his bijective proof of the Young tableau#Dimension of a representation, hook-length formula for the number of Young tableaux, and his work on random walks. He was a keynote speaker alongside George Andrews (mathematician), George Andrews and Doron Zeilberger at the 2006 Harvey Mudd College Mathematics Conference on Enumerative Combinatorics. Pak is an Associate Editor for the journal Discrete Mathematics (journal), ''Discrete Mathematics''. He gave a László Fejes Tóth, Fejes Tóth Lecture at the University of Calgary in February 2009. In 2018, he was an List of International Congresses of Mathematicians Plenary and Invited Speakers#2018, Rio de Janeiro, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doron Zeilberger
Doron Zeilberger (דורון ציילברגר, born 2 July 1950 in Haifa, Israel) is an Israeli mathematician, known for his work in combinatorics. Education and career He received his doctorate from the Weizmann Institute of Science in 1976, under the direction of Harry Dym, with the thesis "New Approaches and Results in the Theory of Discrete Analytic Functions." He is a Board of Governors Professor of Mathematics at Rutgers University. Contributions Zeilberger has made contributions to combinatorics, hypergeometric identities, and q-series. Zeilberger gave the first proof of the alternating sign matrix conjecture, noteworthy not only for its mathematical content, but also for the fact that Zeilberger recruited nearly a hundred volunteer checkers to "pre-referee" the paper. In 2011, together with Manuel Kauers and Christoph Koutschan, Zeilberger proved the ''q''-TSPP conjecture, which was independently stated in 1983 by George Andrews and David P. Robbins. Zeilberger is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Herbert S
Herbert may refer to: People Individuals * Herbert (musician), a pseudonym of Matthew Herbert Name * Herbert (given name) * Herbert (surname) Places Antarctica * Herbert Mountains, Coats Land * Herbert Sound, Graham Land Australia * Herbert, Northern Territory, a rural locality * Herbert, South Australia. former government town * Division of Herbert, an electoral district in Queensland * Herbert River, a river in Queensland * County of Herbert, a cadastral unit in South Australia Canada * Herbert, Saskatchewan, Canada, a town * Herbert Road, St. Albert, Canada New Zealand * Herbert, New Zealand, a town * Mount Herbert (New Zealand) United States * Herbert, Illinois, an unincorporated community * Herbert, Michigan, a former settlement * Herbert Creek, a stream in South Dakota * Herbert Island, Alaska Arts, entertainment, and media Fictional entities * Herbert (Disney character) * Herbert Pocket (''Great Expectations'' character), Pip's close friend and roommate in the Cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert Nijenhuis
Albert Nijenhuis (November 21, 1926 – February 13, 2015) was a Dutch-American mathematician who specialized in differential geometry and the theory of deformations in algebra and geometry, and later worked in combinatorics. His high school studies at the gymnasium in Arnhem were interrupted by the evacuation of Arnhem by the Nazis after the failure of Operation Market Garden by the Allies. He continued his high school mathematical studies by himself on his grandparents’ farm, and then took state exams in 1945. His university studies were carried out at the University of Amsterdam, where he received the degree of Candidaat (equivalent to a Bachelor of Science) in 1947, and a Doctorandus (equivalent to a Masters in Science) in 1950, cum laude. He was a Medewerker (associate) at the Mathematisch Centrum (now the Centrum Wiskunde & Informatica) in Amsterdam 1951–1952. He obtained a Ph.D. in mathematics in 1952, cum laude (Theory of the geometric object). His thesis advisor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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