Putnam 2005 B6

Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n$. For $\pi \in S_n$, let $\sigma(\pi) = 1$ if $\pi$ is an even permutation and $\sigma(\pi) = -1$ if $\pi$ is an odd permutation. Also, let $\nu(\pi)$ denote the number of fixed points of $\pi$. Show that $$\sum_{\pi \in S_n} \frac{\sigma(\pi)}{\nu(\pi) + 1} = (-1)^{n+1} \frac{n}{n+1}.$$