Transposition/Sign/Fact/Proof

Suppose that the transposition ${\displaystyle {}\tau }$ swaps the numbers ${\displaystyle {}k<\ell }$. In case ${\displaystyle {}k+1<\ell }$, let ${\displaystyle {}\rho }$ denote the transposition of the neighbors ${\displaystyle {}k}$ and ${\displaystyle {}k+1}$, and let ${\displaystyle {}{\tilde {\tau }}}$ denote the transposition of ${\displaystyle {}k+1}$ and ${\displaystyle {}\ell }$. Then we have the relationship

${\displaystyle {}\tau =\rho \circ {\tilde {\tau }}\circ \rho \,,}$

which can be checked for the relevant elements ${\displaystyle {}k,k+1,\ell }$ directly. Due to the homomorphism property, we have ${\displaystyle {}\operatorname {sgn} (\tau )=\operatorname {sgn} ({\tilde {\tau }})}$. Therefore, it is enough to prove the statement fur such transpositions that swap two neighbors. Such a transposition has only one inversion, and the claim follows from fact.