Saturday, 2 November 2013

Permutations

Consider the element   a  ,  b  ,  c   of a set   {  a ,  b ,  c  }  .
We can arrange these letters   a ,  b ,  c ,  in the following six manners .

a , b, c ;  a , c , b ;  b , c , a  ;   c , a , b  ;  c , b , a  ;

i.e. there are 3  !  =  6   ways of arranging them or there are   3  !    permutations of the three elements   a ,  b ,  c  .   The permutations are written as  P1   P2  ............. ,  and we adopt a two line notations to express the permutations .

In the first line we write the element in their natural order and in the line below it we write them in the order in which they have been arranged .

i. e.         P1  =  (  a  b  c  )
(  a  b  c  )

i.e.   a    ---->   a     ,   b  ---->  b   ,     c  ----->  c     ,

There is no change and this type of permutation is called   Identity  Permutation  and is written as  l   .

We can also understand Permutations as follows  ---::

Permutation is used with several slightly different meanings, all related to the act of permuting (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). For example, an anagram of a word is a permutation of its letters. The study of permutations in this sense generally belongs to the field of combinatorics.
The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×⋯×1, which is commonly denoted as "n factorial" and written "n!".
Permutations occur, in more or less prominent ways, in almost every domain of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of sorting algorithms in computer science.
In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself (i.e., a map SS for which every element of S occurs exactly once as image value). This is related to the rearrangement of S in which each element s takes the place of the corresponding f(s). The collection of such permutations form a symmetric group. The key to its structure is the possibility to compose permutations: performing two given rearrangements in succession defines a third rearrangement, the composition. Permutations may act on composite objects by rearranging their components, or by certain replacements (substitutions) of symbols.
In elementary combinatorics, the k-permutations, or partial permutations, are the sequences of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.