Parity of a Permutation Determine the parity of the following.
$$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 7 & 4 & 1 & 6 & 5 & 3 \end{pmatrix}$$
Workings:
$\sigma = \begin{pmatrix} 1 & 2 & 7 & 3 & 4 \end{pmatrix}\begin{pmatrix} 5 & 6 \end{pmatrix}$
$\sigma = \begin{pmatrix} 1 & 4 \end{pmatrix} \begin{pmatrix} 1 & 3 \end{pmatrix} \begin{pmatrix} 1 & 7 \end{pmatrix} \begin{pmatrix} 1 & 2 \end{pmatrix} \begin{pmatrix} 5 & 6 \end{pmatrix}$
The parity is odd.
I'm wondering if this is correct.

If $\tau=\beta_{1}\cdots \beta_{k}$ where each $\beta_{i}$, for $1\le i \le k$, is a transposition, then we have $$sgn(\tau)=(-1)^{k}$$
But, for $\sigma$ we have $k=5$. Hence, $$sgn(\sigma)=(-1)^{5}=-1.$$
