If we have a set of points X in d-dimensional euclidean space, and we look at the set of all n-dimensional hyperplanes that are defined by any subset Y of X (in the sense of being the unique n-dimensional hyperplane containing the points of Y), ranging over all Y in P(X), is there a result for the number of regions defined by this hyperplane arrangement?
edit: I just realized, I'm not sure this makes sense for arbitrary n, only n=d-1.
I will recast the question as how many pieces can $d$ dimensional space be cut into by $n$ $d-1$ dimensional hyperplanes. For $d=2$ it is A000124, $n(n+1)/2+1$. For $d=3$ it is A000125, ${n+1 \choose 3}+n+1$ and the remarks explain why the recurrence is $a(n)=a(n-1)+A000124(n-1)$ The logic carries over to higher dimensions, so that if $a(n,d)$ is the number of regions in $d$ dimensional space created by $n$ hyperplanes, the recurrence is $a(n,d)=a(n-1,d)+a(n-1,d-1)$