Permutationer engelska
In mathematics , a permutation of a set can mean one of two different things:. Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them.
The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science , they are used for analyzing sorting algorithms ; in quantum physics , for describing states of particles; and in biology , for describing RNA sequences.
Permutationer
The number of permutations of n distinct objects is n factorial , usually written as n! According to the second meaning, a permutation of a set S is defined as a bijection from S to itself. The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition of functions performing one rearrangement after the other , which results in another function rearrangement.
permutation i svenska, översättning, engelska - Glosbe ordbok
In elementary combinatorics, the k -permutations , or partial permutations , are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations in the previous sense. In Greece, Plutarch wrote that Xenocrates of Chalcedon — BC discovered the number of different syllables possible in the Greek language.
This would have been the first attempt on record to solve a difficult problem in permutations and combinations. Al-Khalil — , an Arab mathematician and cryptographer , wrote the Book of Cryptographic Messages.
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It contains the first use of permutations and combinations, to list all possible Arabic words with and without vowels. The rule to determine the number of permutations of n objects was known in Indian culture around AD. The product of multiplication of the arithmetical series beginning and increasing by unity and continued to the number of places, will be the variations of number with specific figures.
In , Fabian Stedman described factorials when explaining the number of permutations of bells in change ringing. Starting from two bells: "first, two must be admitted to be varied in two ways", which he illustrates by showing 1 2 and 2 1. His explanation involves "cast away 3, and 1. Effectively, this is a recursive process. He continues with five bells using the "casting away" method and tabulates the resulting combinations.
Now the nature of these methods is such, that the changes on one number comprehends the changes on all lesser numbers, Stedman widens the consideration of permutations; he goes on to consider the number of permutations of the letters of the alphabet and of horses from a stable of A first case in which seemingly unrelated mathematical questions were studied with the help of permutations occurred around , when Joseph Louis Lagrange , in the study of polynomial equations, observed that properties of the permutations of the roots of an equation are related to the possibilities to solve it.
This line of work ultimately resulted, through the work of Évariste Galois , in Galois theory , which gives a complete description of what is possible and impossible with respect to solving polynomial equations in one unknown by radicals. In modern mathematics, there are many similar situations in which understanding a problem requires studying certain permutations related to it.
Permutations played an important role in the cryptanalysis of the Enigma machine , a cipher device used by Nazi Germany during World War II. In particular, one important property of permutations, namely, that two permutations are conjugate exactly when they have the same cycle type, was used by cryptologist Marian Rejewski to break the German Enigma cipher in turn of years In mathematics texts it is customary to denote permutations using lowercase Greek letters.
A permutation can be defined as a bijection an invertible mapping, a one-to-one and onto function from a set S to itself:. Composition is usually written without a dot or other sign. As a bijection from a set to itself, a permutation is a function that performs a rearrangement of a set, termed an active permutation or substitution. An older viewpoint sees a permutation as an ordered arrangement or list of all the elements of S , called a passive permutation.
This meaning is subtly distinct from how passive i. A cycle consisting of k elements is called a k -cycle. See § Cycle notation below.