Permutation/Sign/Product of differences/Definition

Let ${\displaystyle {}M=\{1,\ldots ,n\}}$, and let ${\displaystyle {}\pi }$ be a permutation on ${\displaystyle {}M}$. Then we call the number

${\displaystyle {}\operatorname {sgn} (\pi )=\prod _{i<j}{\frac {\pi (j)-\pi (i)}{j-i}}\,}$

the sign of the permutation ${\displaystyle {}\pi }$.