| 释义 |
permutation
per·mu·ta·tion P0201800 (pûr′myo͝o-tā′shən)n.1. a. The process of altering the order of a given set of objects in a group.b. The result of such a process; a rearrangement or recombination of elements: permutations of gene order.c. Mathematics A rearrangement of the elements of a set.2. A complete change; a transformation: the country's permutation into a modern democracy.
per′mu·ta′tion·al adj.permutation (ˌpɜːmjʊˈteɪʃən) n1. (Mathematics) maths a. an ordered arrangement of the numbers, terms, etc, of a set into specified groups: the permutations of a, b, and c, taken two at a time, are ab, ba, ac, ca, bc, cb. b. a group formed in this way. The number of permutations of n objects taken r at a time is n!/(n–r)!. Symbol: nPr Compare combination62. a combination of items made by reordering3. an alteration; transformation4. (Gambling, except Cards) a fixed combination for selections of results on football pools. Usually shortened to: perm [C14: from Latin permūtātiō, from permūtāre to change thoroughly; see mutation] ˌpermuˈtational adjper•mu•ta•tion (ˌpɜr myʊˈteɪ ʃən)
n. 1. the act of permuting or permutating; alteration; transformation. 2. Math. a. the act of changing the order of set elements arranged in a particular way, as abc into acb or bac. b. any of the resulting arrangements. Compare combination (def. 8). [1325–75; Middle English permutacioun (< Anglo-French, Middle French) < Latin permūtātiō= permūtā(re) to permute + -tiō -tion] per`mu•ta′tion•al, adj. per`mu•ta′tion•ist, n. Thesaurus| Noun | 1. |  permutation - an event in which one thing is substituted for another; "the replacement of lost blood by a transfusion of donor blood"substitution, switch, transposition, replacementvariation, fluctuation - an instance of change; the rate or magnitude of change | | 2. | permutation - the act of changing the arrangement of a given number of elementsmathematical operation, mathematical process, operation - (mathematics) calculation by mathematical methods; "the problems at the end of the chapter demonstrated the mathematical processes involved in the derivation"; "they were learning the basic operations of arithmetic" | | 3. | permutation - complete change in character or condition; "the permutations...taking place in the physical world"- Henry Millertranslation, transformation - the act of changing in form or shape or appearance; "a photograph is a translation of a scene onto a two-dimensional surface" | | 4. | permutation - act of changing the lineal order of objects in a groupreordering - a rearrangement in a different order |
permutationnoun transformation, change, shift, variation, modification, alteration, mutation, transmutation, transposition an infinite variety of permutationspermutationnounThe process or result of making or becoming different:alteration, change, modification, mutation, variation.Translationspermutation (pəːmjuˈteiʃən) noun a particular order in which things are arranged. We can write down these numbers in various permutations. 排列 排列
permutation
permutation1. Mathsa. an ordered arrangement of the numbers, terms, etc., of a set into specified groups b. a group formed in this way. The number of permutations of n objects taken r at a time is n&! / (n--r)&! 2. a fixed combination for selections of results on football pools Permutation the replacement of each of the elements a of a given set by another element ϕ(a) of the same set. Each element of the initial set must be obtained precisely once as a result of the permutation. Thus, a permutation is essentially a one-to-one mapping of a set onto itself. The concept of permutation is applied chiefly to finite sets, and only this case will be considered here. A permutation is commonly symbolized 
Here, each element of the given set is written above the element corresponding to it. Since the properties of a permutation are independent of the nature of the elements a, b, …, c, the latter are usually replaced by the integers 1, 2, …, n. In the upper row, the integers are usually written in their natural order. The permutation then has the form 
or simply 
where ϕ1, ϕ2, …, ϕn are the numbers 1, 2, …, n, possibly in a different order. Thus, the second row of a permutation is an arrangement, ϕ1, ϕ2, …, ϕn of the numbers 1, 2, …, n. There are as many different permutations of n elements as there are arrangements—that is, n! = 1 × 2 × 3 × … × n. The permutation 
which leaves invariant all elements, is called the identity permutation. Every permutation A has an inverse, that is, a permutation that carries ϕi to i. The inverse of A is denoted by A–1. For example, if 
then 
The result of the successive application of two permutations A and B is itself a permutation C. If A carries i to ϕi and B carries ϕi to ψi, then C carries i to ψi. C is called the product of A and B; this relationship is written C = AB. For example, if 
and 
then 
Multiplication of permutations is not commutative; that is, in general AB ≠ BA. In the above example, 
It can be easily seen that IA = AI = A, that AA–1 = A–1A = I, and that the associative law A(BC) = (AB)C holds. Thus, all the permutations of n elements form a group, which is called the symmetric group of degree n. A permutation that interchanges two elements i and j is called a transposition and is denoted by (i, j); for example, 
Any permutation can be factored into a product of transpositions. When a given permutation is factored into a product of transpositions in different ways, there will be either an even or an odd number of factors. The permutation will accordingly be said to be even or odd; for example, A = (1, 3) (5, 4) (5, 1) is an odd permutation. Define an inversion as an ordered pair of natural numbers such that the first is greater than the second. It turns out that the parity of a permutation can also be determined from the number of inversions in the lower row of the permutation if the numbers in the upper row are arranged in their natural order. The parity of the permutation coincides with the parity of the number of inversions. For example, the lower row of A contains five inversions: (3, 2), (3, 1), (2, 1), (5, 1), and (5, 4). There exist n!/2 even and n!/2 odd permutations of n elements. A permutation that cyclically permutes a given group of elements while leaving invariant the other elements is called a cycle. The number of permuted elements is called the length of the cycle. For example, A is a cycle of length four: it carries 1 to 3, 3 to 5, 5 to 4, and 4 to 1. This fact is often denoted simply by A = (1, 3, 5, 4). A transposition is a cycle of length two. Any permutation can be factored into a product of disjoint cycles, that is, a product of cycles without common elements. For example, 
REFERENCEKurosh, A. G. Kurs vysshei algebry, 10th ed. Moscow-Leningrad, 1971.permutation[‚pər·myə′tā·shən] (mathematics) A function which rearranges a finite number of symbols; more precisely, a one-to-one function of a finite set onto itself. permutation (mathematics)1. An ordering of a certain number of elementsof a given set.
For instance, the permutations of (1,2,3) are (1,2,3) (2,3,1)(3,1,2) (3,2,1) (1,3,2) (2,1,3).
Permutations form one of the canonical examples of a "group"- they can be composed and you can find an inverse permutationthat reverses the action of any given permutation.
The number of permutations of r things taken from a set of nis
n P r = n! / (n-r)!
where "n P r" is usually written with n and r as subscriptsand n! is the factorial of n.
What the football pools call a "permutation" is not apermutation but a combination - the order does not matter.
2. A bijection for which the domain and range are thesame set and so
f(f'(x)) = f'(f(x)) = x.permutationOne possible combination of items out of a larger set of items. For example, with the set of numbers 1, 2 and 3, there are six possible permutations: 12, 21, 13, 31, 23 and 32.permutation
permutation (pĕr″mū-tā′shŭn) [L. per, completely, + mutare, to change] Transformation; complete change; act of altering objects in a group.Permutation
Related to Permutation: Permutation matrix, Permutation and CombinationPERMUTATION, civil law. Exchange; barter.
2. This contract is formed by the consent of the parties, but delivery is indispensable; for, without it, it mere agreement. Dig. 31, 77, 4; Code, 4, 64, 3.
3. Permutation differs from sale in this, that in the former a delivery of the articles sold must be made, while in the latter it is unnecessary. It agrees with the contract of sale, however, in the following particulars: 1. That he to whom the delivery is made acquires the right or faculty of prescribing. Dig. 41, 3, 4, 17. 2. That the contracting parties are bound to guaranty to each other the title of the things delivered. Code, 4, 64, 1. 3. That they are bound to take back the things delivered, when they have latent defects which they have concealed. Dig. 21, 1, 63. See Aso & Man. Inst. B. 2, t. 16, c. 1; Nutation; Transfer. FinancialSeeNPRAcronymsSeePpermutation
Related to permutation: Permutation matrix, Permutation and CombinationSynonyms for permutationnoun transformationSynonyms- transformation
- change
- shift
- variation
- modification
- alteration
- mutation
- transmutation
- transposition
Synonyms for permutationnoun the process or result of making or becoming differentSynonyms- alteration
- change
- modification
- mutation
- variation
Synonyms for permutationnoun an event in which one thing is substituted for anotherSynonyms- substitution
- switch
- transposition
- replacement
Related Wordsnoun the act of changing the arrangement of a given number of elementsRelated Words- mathematical operation
- mathematical process
- operation
noun complete change in character or conditionRelated Words- translation
- transformation
noun act of changing the lineal order of objects in a groupRelated Words |