<s>[INST] Permutation in discrete math Is the permutation 
$$\begin{pmatrix} 1& 2 &3 &4 &5 &6&7 \\ 7 & 4 & 2 & 1 & 3 & 6 & 5 \end{pmatrix}$$
even or odd?
The product of disjoint cycles is $$\begin{pmatrix}1& 7&5&3&2&4\end{pmatrix}\begin{pmatrix}6\end{pmatrix}$$
and the transposition are $$\begin{pmatrix}1 & 7 \end{pmatrix}\begin{pmatrix}1 & 5 \end{pmatrix}\begin{pmatrix}1 & 3\end{pmatrix}\begin{pmatrix}1 & 2 \end{pmatrix}\begin{pmatrix}1 & 4 \end{pmatrix}\begin{pmatrix}6 \end{pmatrix}$$ is it correct?
and the answer is even??
I feel confused about this..
 [/INST] You only need the decomposition in disjoint cycles. Since $\;(1\;7\;5\;3\;2\;4)\;$ as even length, it is an odd cycle and thus is your original permutation
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