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The rule to determine the number of permutations of n objects was known in Indian ... A k-permutation of a multiset M is a sequence of k elements of M in which each ...
Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ , or less commonly the Latin lower case e .
In combinatorics, the Eulerian number (,) is the number of permutations of the numbers 1 to in which exactly elements are greater than the previous element (permutations with "ascents"). Leonhard Euler investigated them and associated polynomials in his 1755 book Institutiones calculi differentialis .
Multinomial coefficient as a product of binomial coefficients, counting the permutations of the letters of MISSISSIPPI. The multinomial coefficient (, …,) is also the number of distinct ways to permute a multiset of n elements, where k i is the multiplicity of each of the i th element. For example, the number of distinct permutations of the ...
In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. [1] [2] In some cases, cyclic permutations are referred to as cycles; [3] if a cyclic permutation has k elements, it may be called a k-cycle. Some authors widen this definition to include permutations with fixed points in ...
Every permutation can be produced by a sequence of transpositions (2-element exchanges). Let the following be one such decomposition σ = T 1 T 2... T k. We want to show that the parity of k is equal to the parity of the number of inversions of σ. Every transposition can be written as a product of an odd number of transpositions of adjacent ...
[1]: 322 If the integer k is one of them, then the standard basis vector e k is an eigenvector of P. [1]: 118 To calculate the complex eigenvalues of P, write the permutation as a composition of disjoint cycles, say =. (Permutations of disjoint subsets commute, so it doesn't matter here whether we are composing right-to-left or left-to-right.)
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number.