Statistics - Odd and Even Permutation


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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:

${ \{ 2, 1, 3, 4 \} \\[7pt] \{ 1, 3, 2, 4 \} \\[7pt] \{ 1, 2, 4, 3 \} \\[7pt] \{ 3, 2, 1, 4 \} \\[7pt] \{ 4, 2, 3, 1 \} \\[7pt] \{ 1, 4, 3, 2 \} }$

Step 2:

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

${ \{ 2, 3, 4, 1 \} \\[7pt] \{ 2, 4, 1, 3 \} \\[7pt] \{ 3, 1, 4, 2 \} \\[7pt] \{ 3, 4, 2, 1 \} \\[7pt] \{ 4, 1, 2, 3 \} \\[7pt] \{ 4, 3, 1, 2 \} }$

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:

${ \{ 1, 2, 3, 4 \} }$

Step 2:

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

${ \{ 1, 3, 4, 2 \} \\[7pt] \{ 1, 4, 2, 3 \} \\[7pt] \{ 2, 1, 4, 3 \} \\[7pt] \{ 2, 3, 1, 4 \} \\[7pt] \{ 2, 4, 3, 1 \} \\[7pt] \{ 3, 1, 2, 4 \} \\[7pt] \{ 3, 2, 4, 1 \} \\[7pt] \{ 3, 4, 1, 2 \} \\[7pt] \{ 4, 1, 3, 2 \} \\[7pt] \{ 4, 2, 1, 3 \} \\[7pt] \{ 4, 3, 2, 1 \} }$

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