Permutation

In mathematics, a permutation is a bijection of a set onto itself. It can be interpreted as a new order of the members of a sequence or linear order, or as the act or process of changing the linear order of an ordered set. An example of the first interpretation is the six permutations of the set {1, 2, 3}, which are the six 3-tuples (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). They correspond to the six bijections that map 1, 2, 3, to the first, the second and the third member of the tuple, respectively. 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. The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n. According to the second meaning, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value. Such a function σ : S → S {\displaystyle \sigma :S\to S} is equivalent to the rearrangement of the elements of S in which each element i is replaced by the corresponding σ ( i ) {\displaystyle \sigma (i)}...