# Permutation

**PERMUTATION**: On a finite set A of some objects, a permutation π is a one-to-one mapping from A onto intself.

**For example,** consider a set {a, b, c, d}.

A permutation π_{1} = takes a into b, b into d, c into c, and d into a. Alternating, we could write π1(a) = b, π1(b) = d, π1(c) = c, π1(d) = a.

The number of elements in the object set on which a permutation acts is called the degree of the permutation.

For example, the permutation π_{1} = is represented diagrammatically by Figure 6.7 below :

Permutation can be written as (a b d) (c).

The number of edges in a permutation cycle is called the length of the cycle in the permutation.

A permutation π of degree k is said to be of type (σ1, σ2, ...... σk) if π has σi cycles of length i for u = 1, 2, ..... k.

For example, permutation is of type (2, 0, 2, 0, 0, 0, 0, 0).

Clearly, **1σ _{1} 2σ_{2} 3σ_{3} ..... kσ_{k} = k.**

**COMPOSITION OF PERMUTATION**

Consider the two permutations π1 and π2 on an object set {1, 2, 3, 4, 5} : π1 =

and π2 =

.

A composition of these two permutations π2π1 is another permutation obtained by first applying π1 and then applying π2 on the resultant.

That, is

π_{2}π_{1}(1) = π_{2}(2) = 4

π_{2}π_{1}(2) = π_{2}(1) = 3

π_{2}π_{1}(3) = π_{2}(4) = 2

π_{2}π_{1}(4) = π_{2}(5) = 5

π_{2}π_{1}(5) = π_{2}(3) = 1

Thus π2π1 =