Sign of a permutation

Definition

Let $n \geq 2$ and let $\sigma \in S_n$. Write $\sigma = \tau_1 \tau_2 \cdots \tau_m$ as a product of transpositions. The sign of $\sigma$ is $\operatorname{sgn}(\sigma) = (-1)^m$.

Remarks

The parity of $m$ does not depend on the decomposition, so the sign is well defined. Equivalently, $\operatorname{sgn}(\sigma) = (-1)^{N(\sigma)}$, where $N(\sigma)$ is the number of pairs $i < j$ with $\sigma(i) > \sigma(j)$. The sign of a permutation is the determinant of its permutation matrix.

Explanations

L1 human

Each permutation is a product of swaps, and the number of swaps is always even or always odd. The sign records this: $+1$ for an even number and $-1$ for an odd number. A single swap has sign $-1$.