How many column switches have occurred?

Question

If I look at the permutation $[1,2,...,n] \to [2,3,...,n,1]$, how many column switches have occurred?

I'm pretty sure it's n switches, and so the corresponding permutation matrix has determinant =$ (-1)^n.$

The solution gives $ (-1)^{n+1}. $ It's probably a typo, or I may have possibly overlooked something.

Thanks,

Answer 1

Let $(a,b)$ denote the switch of $a$ and $b$. Note that you can permute $n$ elements with the following sequence of switches (in order from left to right): $$ (1,2),(2,3),\dots,(n-1,n) $$ In total, that's $n-1$ switches. The solution is then $(-1)^{n-1} = (-1)^{n+1}$, so the given solution is correct.

Answer 2

Your permutation, in cycles notation, is the cycle $(1\,2\,\dots\, n)$ The signature of a cycle of length $\ell$ is $(-1)^{\ell-1}$, whence the answer.