Transposition/Sign/Fact/Proof2

Suppose that the transposition ${\displaystyle {}\tau }$ swaps the numbers ${\displaystyle {}k<\ell }$. We compute the number of inversions of ${\displaystyle {}\tau }$. We determine for every pair ${\displaystyle {}(i,j)}$, ${\displaystyle {}i<j}$, whether it is an inversion. For ${\displaystyle {}i<k}$ we do not have an inversion, since

${\displaystyle {}\tau (i)=i<\min\{k,j\}=\tau (j)\,.}$

The same holds in case ${\displaystyle {}j>\ell }$. So suppose that

${\displaystyle {}k\leq i<j\leq \ell \,.}$

For

${\displaystyle {}k<i<j<\ell \,,}$

we have ${\displaystyle {}\tau (i)=i<j=\tau (j)}$, and there is no inversion. For

${\displaystyle {}k=i<j<\ell \,,}$

we have an inversion, due to

${\displaystyle {}\tau (k)=\ell >j=\tau (j)\,.}$

In the same way we have for

${\displaystyle {}k<i<j=\ell \,}$

an inversion. Moreover, ${\displaystyle {}k<\ell }$ is an inversion. This yields altogether

${\displaystyle {}(\ell -k-1)+(\ell -k-1)+1=2\ell -2k-1\,}$

inversions. Therefore, the number of inversions is odd and the sign of a transposition is ${\displaystyle {}-1}$ due to fact.