SettingsPermutations
Alternating permutation
In combinatorial mathematics, an alternating permutation of the set {1, 2, 3, ..., n} is an arrangement of those numbers into an order c1, ..., cn such that no element ...
In combinatorial mathematics, an alternating permutation of the set {1, 2, 3, ..., n} is an arrangement of those numbers into an order c1, ..., cn such that no element ...
Bender-Knuth involution
In algebraic combinatorics, a Bender–Knuth involution is an involution on the set of semistandard tableaux, introduced by in their study of plane partitions.
In algebraic combinatorics, a Bender–Knuth involution is an involution on the set of semistandard tableaux, introduced by in their study of plane partitions.
Bender–Knuth involution
In algebraic combinatorics, a Bender–Knuth involution is an involution on the set of semistandard tableaux, introduced by in their study of plane partitions.
In algebraic combinatorics, a Bender–Knuth involution is an involution on the set of semistandard tableaux, introduced by in their study of plane partitions.
Bit-reversal permutation
In applied mathematics, a bit-reversal permutation is a permutation of a sequence with n = 2m (a power of two) elements, defined by reversing the binary digits of the ind...
In applied mathematics, a bit-reversal permutation is a permutation of a sequence with n = 2m (a power of two) elements, defined by reversing the binary digits of the ind...
Boustrophedon transform
In mathematics, the boustrophedon transform is a procedure which maps one sequence to another.
In mathematics, the boustrophedon transform is a procedure which maps one sequence to another.
Cayley's theorem
In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G.
In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G.
Change ringing
Change ringing is the art of ringing a set of tuned bells in a series of mathematical patterns called "changes".
Change ringing is the art of ringing a set of tuned bells in a series of mathematical patterns called "changes".
Circular permutation in proteins
Circular permutation is a type of relationship between proteins, whereby the proteins have a changed order of amino acids in their protein sequence, such that the sequence of the first portion o...
Circular permutation is a type of relationship between proteins, whereby the proteins have a changed order of amino acids in their protein sequence, such that the sequence of the first portion o...
Claw-free permutation
In mathematical and computer science field of cryptography, a group of three numbers is said to be a claw of two permutations f0 and f1 if :f0 = f1 = z.
In mathematical and computer science field of cryptography, a group of three numbers is said to be a claw of two permutations f0 and f1 if :f0 = f1 = z.
Computing the permanent
In mathematics, the computation of the permanent of a matrix is a problem that is believed to be more complex than the computation of the determinant of a matrix despite the apparent similarity ...
In mathematics, the computation of the permanent of a matrix is a problem that is believed to be more complex than the computation of the determinant of a matrix despite the apparent similarity ...
Costas array
In mathematics, a Costas array can be regarded geometrically as a set of n points lying on the squares of a n×n checkerboard, such that each row or column contains only one poi...
In mathematics, a Costas array can be regarded geometrically as a set of n points lying on the squares of a n×n checkerboard, such that each row or column contains only one poi...
Cycle (mathematics)
In mathematics, and in particular in group theory, a cycle is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fash...
In mathematics, and in particular in group theory, a cycle is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fash...
Cycle notation
In combinatorial mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles.
In combinatorial mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles.
Cycles and fixed points
In combinatorial mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S.
In combinatorial mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S.
Cyclic number
A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number.
A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number.
Cyclic permutation
A cyclic permutation or circular permutation is a permutation built from one or more sets of elements in cyclic order.
A cyclic permutation or circular permutation is a permutation built from one or more sets of elements in cyclic order.
Derangement
In combinatorial mathematics, a derangement is a permutation of the elements of a set such that none of the elements appear in their original position.
In combinatorial mathematics, a derangement is a permutation of the elements of a set such that none of the elements appear in their original position.
Fifteen puzzle
The 15-puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others) is a sliding puzzle that consists of a frame of numbered square tiles ...
The 15-puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others) is a sliding puzzle that consists of a frame of numbered square tiles ...
Fisher-Yates shuffle
The Fisher–Yates shuffle (named after Ronald Fisher and Frank Yates), also known as the Knuth shuffle (after Donald Knuth), is an algorithm for generating a random permutation of a finite ...
The Fisher–Yates shuffle (named after Ronald Fisher and Frank Yates), also known as the Knuth shuffle (after Donald Knuth), is an algorithm for generating a random permutation of a finite ...
Fisher–Yates shuffle
The Fisher–Yates shuffle (named after Ronald Fisher and Frank Yates), also known as the Knuth shuffle (after Donald Knuth), is an algorithm for generating a random permutation of a finite ...
The Fisher–Yates shuffle (named after Ronald Fisher and Frank Yates), also known as the Knuth shuffle (after Donald Knuth), is an algorithm for generating a random permutation of a finite ...
Generalized permutation matrix
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row...
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row...
Ghost Leg
Ghost Leg (Chinese: 畫鬼腳), known in Japan as Amidakuji (阿弥陀籤) or in Korea as 사다리타기 (literally translated as "ladder climing"), is a method of lottery designed to create random pairing...
Ghost Leg (Chinese: 畫鬼腳), known in Japan as Amidakuji (阿弥陀籤) or in Korea as 사다리타기 (literally translated as "ladder climing"), is a method of lottery designed to create random pairing...
Immanant of a matrix
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent.
In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent.
In-place matrix transposition
In-place matrix transposition, also called in-situ matrix transposition, is the problem of transposing an N×M matrix in-place in computer memory, ideally with O(1) (bounded) ad...
In-place matrix transposition, also called in-situ matrix transposition, is the problem of transposing an N×M matrix in-place in computer memory, ideally with O(1) (bounded) ad...
Inversion (discrete mathematics)
In computer science and discrete mathematics, an inversion in a sequence of numbers is a pair of numbers in the sequence that are "out of order" with respect to an ascending or descending order.
In computer science and discrete mathematics, an inversion in a sequence of numbers is a pair of numbers in the sequence that are "out of order" with respect to an ascending or descending order.
Josephus problem
In computer science and mathematics, the Josephus Problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game.
In computer science and mathematics, the Josephus Problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game.
Landau's function
In mathematics, Landau's function g, named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn.
In mathematics, Landau's function g, named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric group Sn.
Lehmer code
In mathematics and in particular in combinatorics, the Lehmer code is a particular way to encode each possible permutation of a sequence of n numbers.
In mathematics and in particular in combinatorics, the Lehmer code is a particular way to encode each possible permutation of a sequence of n numbers.
Levi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus.
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus.
Mantel test
The Mantel test, named after Nathan Mantel, is a statistical test of the correlation between two matrices.
The Mantel test, named after Nathan Mantel, is a statistical test of the correlation between two matrices.
Method ringing
Method ringing (also known as scientific ringing) is a form of change ringing (the practice of ringing a series of mathematical permutations on tuned bells, rather than a melody).
Method ringing (also known as scientific ringing) is a form of change ringing (the practice of ringing a series of mathematical permutations on tuned bells, rather than a melody).
Ménage problem
In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of heterosexual couples at a dining tab...
In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of heterosexual couples at a dining tab...
Narayana number
In combinatorics, the Narayana numbers N(n, k), n = 1, 2, 3 ..., 1 ≤ k ≤ n, form a triangular array of natural numbers, called Narayana ...
In combinatorics, the Narayana numbers N(n, k), n = 1, 2, 3 ..., 1 ≤ k ≤ n, form a triangular array of natural numbers, called Narayana ...
Order statistic
In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value.
In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value.
Parastatistics
In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models.
In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models.
Parity of a permutation
The bijective mappings from X to X) fall into two classes of equal size: the even permutations and the odd permutations.
The bijective mappings from X to X) fall into two classes of equal size: the even permutations and the odd permutations.
Permanent
The permanent of a square matrix in linear algebra, is a function of the matrix similar to the determinant.
The permanent of a square matrix in linear algebra, is a function of the matrix similar to the determinant.
Permutable prime
A permutable prime is a prime number, which, in a given base, can have its digits switched to any possible permutation and still spell a prime number.
A permutable prime is a prime number, which, in a given base, can have its digits switched to any possible permutation and still spell a prime number.
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values.
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values.
Permutation (music)
In music, a permutation of a set is a transformation of its prime form by applying zero or more of certain operations, specifically transposition, inversion, and retrograde.
In music, a permutation of a set is a transformation of its prime form by applying zero or more of certain operations, specifically transposition, inversion, and retrograde.
Permutation automaton
A permutation automaton (or p-automaton) is an automaton such that each input permutes the set of states.
A permutation automaton (or p-automaton) is an automaton such that each input permutes the set of states.
Permutation cipher
In classical cryptography, a permutation cipher is a transposition cipher in which the key is a permutation.
In classical cryptography, a permutation cipher is a transposition cipher in which the key is a permutation.
Permutation matrix
In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0's elsewhere.
In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0's elsewhere.
Permutation pattern
In combinatorial mathematics and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation.
In combinatorial mathematics and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation.
Permutohedron
In mathematics, the permutohedron of order n (also spelled permutahedron) is an (n − 1)-dimensional polytope embedded in an n-dimensional space, the vertices of...
In mathematics, the permutohedron of order n (also spelled permutahedron) is an (n − 1)-dimensional polytope embedded in an n-dimensional space, the vertices of...
Random permutation
A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable.
A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable.
Rencontres numbers
In combinatorial mathematics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fix...
In combinatorial mathematics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fix...
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.
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.
Riemann series theorem
In mathematics, the Riemann series theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arr...
In mathematics, the Riemann series theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arr...
Ring of symmetric functions
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity.
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity.
Robinson-Schensted correspondence
In mathematics, the Robinson–Schensted correspondence is a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape.
In mathematics, the Robinson–Schensted correspondence is a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape.
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-nega...
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-nega...
Robinson–Schensted correspondence
In mathematics, the Robinson–Schensted correspondence is a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape.
In mathematics, the Robinson–Schensted correspondence is a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape.
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 generalized permutation...
In mathematics, the Robinson–Schensted–Knuth correspondence, also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between generalized permutation...
Rook polynomial
In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two roo...
In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two roo...
Separable permutation
In combinatorial mathematics, a separable permutation is a permutation that can be obtained from the trivial permutation 1 by direct sums and skew sums.
In combinatorial mathematics, a separable permutation is a permutation that can be obtained from the trivial permutation 1 by direct sums and skew sums.
Shuffling
Shuffling is a procedure used by Lou Skuntz to randomize a deck of playing cards to provide an element of chance in card games.
Shuffling is a procedure used by Lou Skuntz to randomize a deck of playing cards to provide an element of chance in card games.
Stanley-Wilf conjecture
The Stanley–Wilf conjecture, formulated independently by Richard P. Stanley and Herbert Wilf in the late 1980s, was a conjecture in permutation patterns until it was resolved by Adam Marcus and ...
The Stanley–Wilf conjecture, formulated independently by Richard P. Stanley and Herbert Wilf in the late 1980s, was a conjecture in permutation patterns until it was resolved by Adam Marcus and ...
Stanley–Wilf conjecture
The Stanley–Wilf conjecture, formulated independently by Richard P. Stanley and Herbert Wilf in the late 1980s, was a conjecture in permutation patterns until it was resolved by Adam Marcus and ...
The Stanley–Wilf conjecture, formulated independently by Richard P. Stanley and Herbert Wilf in the late 1980s, was a conjecture in permutation patterns until it was resolved by Adam Marcus and ...
Steinhaus-Johnson-Trotter algorithm
The Steinhaus–Johnson–Trotter algorithm or Johnson–Trotter algorithm is an algorithm that generates permutations by transposing elements.
The Steinhaus–Johnson–Trotter algorithm or Johnson–Trotter algorithm is an algorithm that generates permutations by transposing elements.
Steinhaus-Johnson–Trotter algorithm
The Steinhaus–Johnson–Trotter algorithm or Johnson–Trotter algorithm, also called plain changes, is an algorithm that generates all of the permutations of n elements.
The Steinhaus–Johnson–Trotter algorithm or Johnson–Trotter algorithm, also called plain changes, is an algorithm that generates all of the permutations of n elements.
Stirling number
In mathematics, Stirling numbers arise in a variety of combinatorics problems.
In mathematics, Stirling numbers arise in a variety of combinatorics problems.
Stirling numbers of the first kind
In mathematics, Stirling numbers of the first kind, occur in combinatorics, where they appear in the study of permutations.
In mathematics, Stirling numbers of the first kind, occur in combinatorics, where they appear in the study of permutations.
Substitution-permutation network
In cryptography, an SP-network, or substitution-permutation network (SPN), is a series of linked mathematical operations used in block cipher algorithms such as AES (Rijndael).
In cryptography, an SP-network, or substitution-permutation network (SPN), is a series of linked mathematical operations used in block cipher algorithms such as AES (Rijndael).
Telephone number (mathematics)
In mathematics, the telephone numbers or involution numbers are a sequence of integers that count the number of ways that n subscribers to a telephone system can be linked in pairs, th...
In mathematics, the telephone numbers or involution numbers are a sequence of integers that count the number of ways that n subscribers to a telephone system can be linked in pairs, th...
Transposition (mathematics)
In informal language, a transposition is a function that swaps two elements of a set.
In informal language, a transposition is a function that swaps two elements of a set.
Transposition cipher
In cryptography, a transposition cipher is a method of encryption by which the positions held by units of plaintext (which are commonly characters or groups of characters) are shifted according ...
In cryptography, a transposition cipher is a method of encryption by which the positions held by units of plaintext (which are commonly characters or groups of characters) are shifted according ...
Twelvefold way
In combinatorics, the twelvefold way is a name given to a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of countin...
In combinatorics, the twelvefold way is a name given to a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of countin...
Unpredictable permutation
In combinatorial mathematics, an unpredictable permutation Fk is a permutation where a probabilistic polynomial-time adversary cannot predict the outcome of the challenge query F...
In combinatorial mathematics, an unpredictable permutation Fk is a permutation where a probabilistic polynomial-time adversary cannot predict the outcome of the challenge query F...
Young symmetrizer
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that the image of the element corresponds to an irreducible representati...
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that the image of the element corresponds to an irreducible representati...