Consider the permutations from $S_n$ and let$$K(n,m)=\#\{\pi \in S_n:\exists k,\pi(k)=k,~k\leq m\}$$be the number of such permutations which have a fixed point in the first $m$ positions. I am interested in $N(n,m)=n!-K(n,m).$ By inclusion-exclusion, we have$$N(n,m)=\sum_{i=1}^m (-1)^{i+1}\binom{m}{i}(n-i)!.$$When $m=n,$ in the case of Derangements, we have the approximation$$N(n,n)\sim n!e^{-1}$$to the number of such permutations, obtained by partial summing the Taylor series for the exponential.
There should be a similar approach to approximate $N(n,m)$ but I can't see it. I am particularly interested in $m=n^\theta,$ say $\theta \in (0,1/2),$ as well as $m=\log n,$ perhaps by some optimization of the remainder in the Taylor series for $e^x$ around $x=-1.$