Let \(\sgn\colon S_n\to\theset{\pm1}\) be the sign homomorphism, and let \(\sigma\in S_n\) have disjoint cycle decomposition \(\sigma=c_1\cdots c_t\) with \(c_j\) a cycle of length \(k_j\). Then \[ \sgn(\sigma)=\prod_{j=1}^t(-1)^{k_j-1}. \] In particular, a cycle of length \(k\) is odd if and only if \(k\) is even, and \(\sigma\) is odd if and only if an odd number of the \(c_j\) have even length.
A \(k\)-cycle is a product of \(k-1\) transpositions: \((a_1\,a_2\,\cdots\,a_k)=(a_1\,a_k)(a_1\,a_{k-1})\cdots(a_1\,a_2)\). Since each transposition has sign \(-1\) and \(\sgn\) is a homomorphism, \(\sgn(c_j)=(-1)^{k_j-1}\), and the displayed formula follows by multiplying over \(j\). The factor \((-1)^{k_j-1}\) equals \(-1\) exactly when \(k_j\) is even, so the product equals \(-1\) exactly when the number of even-length cycles is odd.