Permutations

Suppose that you have a set of four distinct objects. A permutation of these objects is a rearrangement of them among themselves. Thus if the objects are red, white, blue, and green counters, then you could permute them by replacing the green counter by the blue one, and vice versa. Or you could replace blue by green, red by blue, and green by red. Both of these are examples of permutations. In practice, it is more convenient to label the four objects 1, 2, 3, and 4. Then the permutation which interchanges blue and green would be “replace 1, 2, 3, 4 by 1, 2, 4, 3.” This permutation is called

1 2 3 4 1 2 4 3

 

 

to show that 4 replaces 3 and 3 replaces 4. Similarly, the permutation which replaces 1, 2, 3, 4 by 3, 2, 4, 1 is called

1 2 3 4 3 2 4 1

 

 

showing that 3 replaces 1, 4 replaces 3, and 1 replaces 4. Note that the original order 1, 2, 3, 4 is not important. The permutation

4 2 1 3 3 2 1 4

 

 

means the same as

1 2 3 4 1 2 4 3

 

  ;

both of them show that 4 replaces 3 and 3 replaces 4. Similarly,

1 2 3 4 3 2 4 1

 

  ,

4 2 1 3 1 2 3 4

 

  ,

2 3 4 1 2 4 1 3

 

  ,

and

2 1 4 3 2 3 1 4

 

 

are identical permutations. If the permutation

1 2 3 4 1 2 4 3

 

 

is followed by

1 2 3 4 3 2 4 1

 

  ,

then the effect is the permutation

1 2 3 4 3 2 1 4

 

  .