Cyclic Shift Minimization

A permutation $P$ - $[P_1, P_2, ..., P_N]$ of size $N$ is an array of size $N$ consisting of integers $1$, $2$, .... $N$ each exactly once, in some order. Define $f(P) = $ number of fixed points, i.e. number of $i$ such that $(P_i = i)$. --- A cyclic shift of $[P_1, P_2, ...., P_N]$ is $[P_{X + 1}, \ldots, P_N, P_1, P_2, ..., P_{X}]$ for each $1 \le X \le N$. Note that there are exactly $N$ cyc