Given a permutation $\sigma=\sigma_1 ... \sigma_k$ (where the $\sigma_i$s are disjoint), is it true to say that its sign is:
$$\operatorname{sign}(\sigma_1,...,\sigma_k)=(-1)^{m_{1}-1}\cdot...\cdot(-1)^{m_{k}-1}$$
where $m_i$ is the length of the cycle $\sigma_i$ for every $1\leq i\leq k$ and if we get $sign(\sigma)=1$ then its even, otherwise its odd?
I tried some examples and they all satisfy this statement.