#
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:

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 } \[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:

###
Step 2:

Swap two numbers two times. Following are the permutations

obtainable:

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 } }$

‘; (vitag.displayInit = window.vitag.displayInit ||

[]).push(function () { viAPItag.display(ad_id); }); }())