Permutation/246531/Inversions and sign/Example

We consider the permutation

${}x$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$
${}\sigma (x)$	${}2$	${}4$	${}6$	${}5$	${}3$	${}1$

with the cycle representation

$\langle 1,2,4,5,3,6\rangle .$

The inversions are

$(1,6),\,(2,5),\,(2,6),\,(3,4),\,(3,5),\,(3,6),\,(4,5),\,(4,6),\,(5,6),$

so there are ${}9$ of those. The sign is ${}(-1)^{9}=-1$ due to fact, and the permutation is odd.