Consider the cyclic permutation operator $\tau$ defined as $$\tau(x_1,...,x_N)=(x_2,...,x_N,x_1)$$ and denote
$$ (\tau f) (x,y) = f(\tau x,\tau y).$$
A cyclically symmetric function is a function $F$ such that $\tau F=F$.
For the Poisson bracket between two cyclically symmetric function we have
$$\sum_{s,s'} \{\tau^s f, \tau^{s'} g \} = \sum_s \tau^s\big(\sum_{s'} \{f,\tau^{s'}g\} \big).$$
I can't figure out the last equality, that is: when do I exploit the cyclically symmetric property? How the partial derivatives change with these permutation?