Statistics Odd and Even Permutation – lesscss

By | June 19, 2019

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:

${ { 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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