Formally,
Count the number of permutations $p = \operatorname{permute}[1, 2, \ldots, n]$ , such that $\forall 1 \le i \le n, |p(i) - i| \ge K$, where $K$ is a given constant.
For example, when $K=1$, the problem is reduced to the normal derangement problem.
Is there a formula for this?