PermutationIn combinatorics, a permutation is a sequence of elements in which no element appears more than once. In a sequence, unlike in a set, the order in which the elements are written down matters. Suppose you have a total of n distinct objects at your disposal and you want to create permutations of k elements selected from those n, where k≤n. In how many ways can that be done?
- We can select the first member of the list in n ways because there are n distinct elements.
- The second member of the list can be filled in (n-1) ways since we have used up one of the n elements already.
- The third member can be filled in (n-2) ways since 2 have been used already.
- This pattern continues until there are k names on the list. This means that the last member can be filled in (n-k+1) ways.
- n * (n-1) * (n-2) * ... * (n-k+1)
- P(n, k) = n! / (n-k)!
There are two main notations for such permutations. In relation notation, one can just arrange the "natural" ordering of the elements being permuted on a row, and the new ordering on another row:
This notation often omits fixed points, that is, elements mapped to themselves; thus (1 3)(2)(4 5) can become simply (1 3)(4 5).
A permutation consists of one cycle is itself called a cycle. The number of entries of a cycle is called the length. For example, the length of (1 2 5) is three.
An identity permutation is the permutation which does not permute anything.
Given a permutation P, the identity permutation I, P's inverse permutation P-1 is the permutation that undoes P, ie., performing P then P-1 is the same as performing the permutation I.
A transposition is a permutation which exchanges two elements and keeps all others fixed. For example (1 3) is a transposition. A transposition is a cycle of length two.
One can define product of two permutations, see Symmetric group and Permutation group. An even permutation is a permutation which can be expressed as a product of even number of transpositions, and the identity permutation is a even permutation as it equals (1 2)(1 2). An odd permutation is an permutation which can be expressed as a product of odd number of transpositions.
A permutation matrix is a matrix representation of permutation.