Exercise 9

For any permutation $\phi$ let $g(\phi)$ be the integer defined in this way.

(This is the product, over all indices $i$ and $j$ with $i<j$, of terms of the given form.)

• Compute the value of $g$ on all $2$-permutations.

• Compute the value of $g$ on all $3$-permutations.

• Prove that $g(\phi)$ is not $0$.

• Prove this.

  $$\operatorname{sgn}(\phi)=\frac{g(\phi)}{\lvert g(\phi)\rvert}$$

Many authors give this formula as the definition of the signum function.