Statistics - Odd and Even Permutation

Consider X as a finite set of at least two elements then permutations of X can be divided into two category of equal size: even permutation and odd permutation.

Odd Permutation

Odd permutation is a set of permutations obtained from odd number of two element swaps in a set. It is denoted by a permutation sumbol of -1. For a set of n numbers where n > 2, there are ${\frac {n!}{2}}$ permutations possible. For example, for n = 1, 2, 3, 4, 5, ..., the odd permutations possible are 0, 1, 3, 12, 60 and so on...

Example

Compute the odd permutation for the following set: {1,2,3,4}.

Solution:

Here n = 4, thus total no. of odd permutation possible are ${\frac {4!}{2} = \frac {24}{2} = 12}$. Following are the steps to generate odd permutations.

Step 1:

Swap two numbers one time. Following are the permutations obtainable:

Step 2:

Swap two numbers three times. Following are the permutations obtainable:

Even Permutation

Even permutation is a set of permutations obtained from even number of two element swaps in a set. It is denoted by a permutation sumbol of +1. For a set of n numbers where n > 2, there are ${\frac {n!}{2}}$ permutations possible. For example, for n = 1, 2, 3, 4, 5, ..., the even permutations possible are 0, 1, 3, 12, 60 and so on...

Example

Compute the even permutation for the following set: {1,2,3,4}.

Solution:

Here n = 4, thus total no. of even permutation possible are ${\frac {4!}{2} = \frac {24}{2} = 12}$. Following are the steps to generate even permutations.

Step 1:

Swap two numbers zero time. Following is the permutation obtainable:

Step 2:

Swap two numbers two times. Following are the permutations obtainable: