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

Permutation in discrete math
Is the permutation
$\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 7& 4& 2& 1& 3& 6& 5\end{array}\right)$
even or odd?
The product of disjoint cycles is
$\left(\begin{array}{cccccc}1& 7& 5& 3& 2& 4\end{array}\right)\left(\begin{array}{c}6\end{array}\right)$
and the transposition are
$\left(\begin{array}{cc}1& 7\end{array}\right)\left(\begin{array}{cc}1& 5\end{array}\right)\left(\begin{array}{cc}1& 3\end{array}\right)\left(\begin{array}{cc}1& 2\end{array}\right)\left(\begin{array}{cc}1& 4\end{array}\right)\left(\begin{array}{c}6\end{array}\right)$
is it correct?
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Abbigail Vaughn
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
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:
$\left\{\left(7,4\right),\left(7,2\right),\left(7,1\right),\left(7,3\right),\left(7,6\right),\left(7,5\right),\left(4,2\right),\left(4,1\right),\left(4,3\right),\left(2,1\right),\left(6,5\right)\right\}$
Step 2
Thus, this is an odd permutation.