“permutation”的概念、定义、翻译、参考文献-科学参考

permutation

Mathematics

A permutation of a set S is a bijection from S to S. The permutations of S form a group, the symmetric group Sym(S) of S, under composition.
In combinatorics, a permutation of n objects is an arrangement n objects. The number of permutations of n objects is equal to n!. The number of ‘permutations of r objects from n objects’ is denoted by ⁿP_r, and equals

$n (n - 1) \dots (n - r + 1) = \frac{n!}{(n - r)!} .$

For example, there are 12 permutations of A, B, C, D taken two at a time: AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC.

Suppose that n objects are of k different kinds, with r₁ alike of one kind, r₂ of a second kind, and so on. Then the number of distinct permutations of the n objects equals the multinomial coefficient

$\frac{n!}{r_{1}! r_{2}! \dots r_{k}!},$

where r₁ + r₂ + … + r_k = n. For example, the number of different anagrams of the word ‘CHEESES’ is 7!/3! 2!, which equals 420. Compare combination.

Statistics

An ordered arrangement of n different objects. The number of permutations (i.e. the number of different possible ordered arrangements) is n!.
For ordered selection of r objects from a set of n(≥r) different objects, the number of permutations of r from n, i.e. the number of different possible ordered selections, is usually denoted by ⁿP_r. In fact, $permutation$ Special values are ⁿP₀=1, ⁿP₁=n, ⁿP_n=n!. For example, the 6 (=3!) permutations of {A, B, C} are ABC, ACB, BAC, BCA, CAB, CBA, and there are 24 (=4 × 3 × 2) permutations of 3 letters from {A, B, C, D}.
If the n objects are not all different, and there are n₁ objects of type 1, n₂ objects of type 2,…, n_k objects of type k, where n₁+n₂+⋯+n_k=n, then the number of different ordered arrangements is $permutation$ For unordered selection, see combination.

Computer

(of a set S) A bijection of S onto itself. When S is finite, a permutation can be portrayed as a rearrangement of the elements of S. The number of permutations of a set of n elements is n!
A permutation of the elements of {1,2,3} can be written
$\begin{array}{l} 1 2 3 \\ 2 1 3 \end{array}$
indicating that 1 is mapped into 2, 2 into 1, and 3 into 3. Alternatively the above can be written, using a cycle notation, as (1 2); this implies that the element 3 is unaltered but that 1 is mapped into 2 and 2 into 1.
For collections of elements in which repeated occurrences of items may exist, a permutation can be described as a rearrangement of elements in which each element appears with the same frequency as before.

Logic

1. The classically valid rule of inference that allows one to permute antecedents in some nested conditional sentences. Where $\to$ is a conditional connective, this is captured by the inference:
$\frac{φ \to (ψ \to ξ)}{ψ \to (φ \to ξ)}$
In some contexts, the term refers to the axiom form of the inference as well, i.e.
- • $(φ \to (ψ \to ξ)) \to (ψ \to (φ \to ξ))$
Because the permutation of antecedents resembles the operation of commutation, this inference is often associated with the commutativity rules for conjunction and disjunction.
2. See exchange.

单词	permutation
释义	permutation Mathematics A permutation of a set S is a bijection from S to S. The permutations of S form a group, the symmetric group Sym(S) of S, under composition. In combinatorics, a permutation of n objects is an arrangement n objects. The number of permutations of n objects is equal to n!. The number of ‘permutations of r objects from n objects’ is denoted by ⁿP_r, and equals $n (n - 1) \dots (n - r + 1) = \frac{n!}{(n - r)!} .$ For example, there are 12 permutations of A, B, C, D taken two at a time: AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC. Suppose that n objects are of k different kinds, with r₁ alike of one kind, r₂ of a second kind, and so on. Then the number of distinct permutations of the n objects equals the multinomial coefficient $\frac{n!}{r_{1}! r_{2}! \dots r_{k}!},$ where r₁ + r₂ + … + r_k = n. For example, the number of different anagrams of the word ‘CHEESES’ is 7!/3! 2!, which equals 420. Compare combination. Statistics An ordered arrangement of n different objects. The number of permutations (i.e. the number of different possible ordered arrangements) is n!. For ordered selection of r objects from a set of n(≥r) different objects, the number of permutations of r from n, i.e. the number of different possible ordered selections, is usually denoted by ⁿP_r. In fact, $permutation$ Special values are ⁿP₀=1, ⁿP₁=n, ⁿP_n=n!. For example, the 6 (=3!) permutations of {A, B, C} are ABC, ACB, BAC, BCA, CAB, CBA, and there are 24 (=4 × 3 × 2) permutations of 3 letters from {A, B, C, D}. If the n objects are not all different, and there are n₁ objects of type 1, n₂ objects of type 2,…, n_k objects of type k, where n₁+n₂+⋯+n_k=n, then the number of different ordered arrangements is $permutation$ For unordered selection, see combination. Computer (of a set S) A bijection of S onto itself. When S is finite, a permutation can be portrayed as a rearrangement of the elements of S. The number of permutations of a set of n elements is n! A permutation of the elements of {1,2,3} can be written $\begin{array}{l} 1 2 3 \\ 2 1 3 \end{array}$ indicating that 1 is mapped into 2, 2 into 1, and 3 into 3. Alternatively the above can be written, using a cycle notation, as (1 2); this implies that the element 3 is unaltered but that 1 is mapped into 2 and 2 into 1. For collections of elements in which repeated occurrences of items may exist, a permutation can be described as a rearrangement of elements in which each element appears with the same frequency as before. Logic 1. The classically valid rule of inference that allows one to permute antecedents in some nested conditional sentences. Where $\to$ is a conditional connective, this is captured by the inference: $\frac{φ \to (ψ \to ξ)}{ψ \to (φ \to ξ)}$ In some contexts, the term refers to the axiom form of the inference as well, i.e. • $(φ \to (ψ \to ξ)) \to (ψ \to (φ \to ξ))$ Because the permutation of antecedents resembles the operation of commutation, this inference is often associated with the commutativity rules for conjunction and disjunction. 2. See exchange.
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