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Schützenberger Involution
In the mathematical field of combinatorics, jeu de taquin is a construction due to which defines an equivalence relation on the set of skew standard Young tableaux. A jeu de taquin slide is a transformation where the numbers in a tableau are moved around in a way similar to how the pieces in the fifteen puzzle move. Two tableaux are jeu de taquin equivalent if one can be transformed into the other via a sequence of such slides. "Jeu de taquin" (literally "teasing game") is the French name for the fifteen puzzle. Definition of a jeu de taquin slide Given a skew standard Young tableau ''T'' of skew shape \lambda / \mu, pick an adjacent empty cell ''c'' that can be added to the skew diagram \lambda\setminus\mu; what this means is that ''c'' must share at least one edge with some cell in ''T'', and \ \cup \lambda\setminus\mu must also be a skew diagram. There are two kinds of slide, depending on whether ''c'' lies to the upper left of ''T'' or to the lower right. Suppose to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definitions A binary relation \,\si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young Tableau
In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schützenberger and Richard P. Stanley. Definitions ''Note: this article uses the English convention for displaying Young diagrams and tableaux''. Diagrams A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-incre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fifteen Puzzle
The 15 puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and more) is a sliding puzzle. It has 15 square tiles numbered 1 to 15 in a frame that is 4 tile positions high and 4 tile positions wide, with one unoccupied position. Tiles in the same row or column of the open position can be moved by sliding them horizontally or vertically, respectively. The goal of the puzzle is to place the tiles in numerical order (from left to right, top to bottom). Named after the number of tiles in the frame, the 15 puzzle may also be called a ''"16 puzzle"'', alluding to its total tile capacity. Similar names are used for different sized variants of the 15 puzzle, such as the 8 puzzle, which has 8 tiles in a 3×3 frame. The ''n'' puzzle is a classical problem for Modeling language, modeling algorithms involving heuristic (computer science), heuristics. Commonly used heuristics for this problem include counting the number of misplaced tiles and finding the sum of the tax ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeu De Taquin
In the mathematical field of combinatorics, jeu de taquin is a construction due to which defines an equivalence relation on the set of skew standard Young tableaux. A jeu de taquin slide is a transformation where the numbers in a tableau are moved around in a way similar to how the pieces in the fifteen puzzle move. Two tableaux are jeu de taquin equivalent if one can be transformed into the other via a sequence of such slides. "Jeu de taquin" (literally "teasing game") is the French name for the fifteen puzzle. Definition of a jeu de taquin slide Given a skew standard Young tableau ''T'' of skew shape \lambda / \mu, pick an adjacent empty cell ''c'' that can be added to the skew diagram \lambda\setminus\mu; what this means is that ''c'' must share at least one edge with some cell in ''T'', and \ \cup \lambda\setminus\mu must also be a skew diagram. There are two kinds of slide, depending on whether ''c'' lies to the upper left of ''T'' or to the lower right. Suppose to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knuth Equivalence
In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableaux. It was discovered by (who called it the tableau algebra), using an operation given by in his study of the longest increasing subsequence of a permutation. It was named the "''monoïde plaxique''" by , who allowed any totally ordered alphabet in the definition. The etymology of the word "''plaxique''" is unclear; it may refer to plate tectonics ("tectonique des plaques" in French), as elementary relations that generate the equivalence allow conditional commutation of generator symbols: they can sometimes slide across each other (in apparent analogy to tectonic plates), but not freely. Definition The plactic monoid over some totally ordered alphabet (often the positive integers) is the monoid with the following presentation: *The generators are the letters of the alphabet *The relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse, : for all in the domain of . Equivalently, applying twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition of two involutions and is an involution if and only if they commute: . Involutions on finite sets The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800: : a_0 = a_1 = 1 and a_n = a_ + (n - 1)a_ for n > 1. The first few terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robinson–Schensted–Knuth Correspondence
In mathematics, the Robinson–Schensted–Knuth correspondence, also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between matrices with non-negative integer entries and pairs of semistandard Young tableaux of equal shape, whose size equals the sum of the entries of . More precisely the weight of is given by the column sums of , and the weight of by its row sums. It is a generalization of the Robinson–Schensted correspondence, in the sense that taking to be a permutation matrix, the pair will be the pair of standard tableaux associated to the permutation under the Robinson–Schensted correspondence. The Robinson–Schensted–Knuth correspondence extends many of the remarkable properties of the Robinson–Schensted correspondence, notably its symmetry: transposition of the matrix results in interchange of the tableaux . The Robinson–Schensted–Knuth correspondence Introduction The Robinson–Schensted correspondence is a bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Littlewood–Richardson Rule
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials. Littlewood–Richardson coefficients depend on three partitions, say \lambda,\mu,\nu, of which \lambda and \mu describe the Schur functions being multiplied, and \nu gives the Schur function of which this is the coefficient in the linear combination; in other wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
