Permutation in discrete math. Is the permutation ((1\ 2\ 3\ 4\ 5\ 6\ 7 ),(7\ 4\ 2\ 1\ 3\ 6\ 5)) even or odd?

chivistaelmore 2022-07-18 Answered
Permutation in discrete math
Is the permutation
( 1 2 3 4 5 6 7 7 4 2 1 3 6 5 )
even or odd?
The product of disjoint cycles is
( 1 7 5 3 2 4 ) ( 6 )
and the transposition are
( 1 7 ) ( 1 5 ) ( 1 3 ) ( 1 2 ) ( 1 4 ) ( 6 )
is it correct?
and the answer is even??
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Answers (2)

Abbigail Vaughn
Answered 2022-07-19 Author has 15 answers
Step 1
This is as you said (1,7,5,3,2,4),
Step 2
and then it is equal to (1,7)(7,5)(5,3)(3,2)(2,4), and hence it is odd.
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Roselyn Daniel
Answered 2022-07-20 Author has 2 answers
Step 1
Another way to check the parity of a permutation is to see how many pairs are out of order (this is the number of inversions). There are 11:
{ ( 7 , 4 ) , ( 7 , 2 ) , ( 7 , 1 ) , ( 7 , 3 ) , ( 7 , 6 ) , ( 7 , 5 ) , ( 4 , 2 ) , ( 4 , 1 ) , ( 4 , 3 ) , ( 2 , 1 ) , ( 6 , 5 ) }
Step 2
Thus, this is an odd permutation.
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