Let $n\in\mathbb{N}$, let $\sigma\in S_n$, and suppose there exist an integer $r\ge 0$ and pairwise disjoint nontrivial cycles $c_1,\dots,c_r\in S_n$ such that $\sigma=c_1c_2\cdots c_r$. For each $i\in\{1,\dots,r\}$, let $k_i\in\{2,\dots,n\}$ denote the length of the cycle $c_i$. If $r=0$, the product $c_1c_2\cdots c_r$ is interpreted as the identity element of $S_n$. Then

\begin{align*} \operatorname{sgn}(\sigma)=\prod_{i=1}^r(-1)^{k_i-1}=(-1)^{\sum_{i=1}^r(k_i-1)}. \end{align*}

For $r=0$, the product is interpreted as the empty product $1$ and the sum as the empty sum $0$.