OLYMPIC-REVENGE 2017 Problem 3

Let $n$ a positive integer. We call a pair $(\pi ,C)$ composed by a permutation $\pi$ $:$ { $1,2,...n$ } $\rightarrow$ { $1,2,...,n$ } and a binary function $C:$ { $1,2,...,n$ } $\rightarrow$ { $0,1$ } "revengeful" if it satisfies the two following conditions: $1)$ For every $i$ $\in$ { $1,2,...,n$ }, there exist $j$ $\in$ $S_{i}=$ { $i, \pi(i),\pi(\pi(i)),...$ } such that $C(j)=1$ . $2)$ If $C(k)=1$ , then $k$ is one of the $v_{2}(|S_{k}|)+1$ highest elements of $S_{k}$ , where $v_{2}(t)$ is the highest nonnegative integer such that $2^{v_{2}(t)}$ divides $t$ , for every positive integer $t$ .

Let $V$ the number of revengeful pairs and $P$ the number of partitions of $n$ with all parts powers of $2$ . Determine $\frac{V}{P}$ .