Let me start off by saying that I'm not a mathematician, so this is probably an easy problem to solve, but I haven't been able to yet..
The problem is that I want to place $n$ objects on a grid with $N$ grid points, and I want to count the number of different permutations this is possible, or at the least an approximate number. However, I am not allowed to place my objects just next to others, and therefore the number of possible places to place them, decreases by a number $s$ after having placed an object.
You can see here, how the situation works for a 4x4 grid with $n=1$ (gives $\binom{4\times 4}{1}=16$ possibilities since no grid points are masked out) and for $n=2$ where one grid point is occupied - removing the option to use its neighboring grid points.
Having placed a single object, the number of grid points available are not $16-1=15$, but rather $16-9=7$. Therefore, the total number of permutations goes from $\binom{16}{2}=16\times 15 / 2 = 120$ to $16\times 7/2=56$. Here I showed it with periodic boundaries as this is preferred, but it is not necessary.
I guess the binomial coefficient $\binom{N}{n}$ is a good starting place, as this can give the number of permutations for a static size grid, however, I haven't been able to figure out how to find it for a non-static $N$.
I thought I could get an approximate answer by just calculating $$ \prod_{i=0}^n N-i\cdot s $$ but this obviously double counts a lot of the same configurations.