Two arrangements of (affine) hyperplanes in $d$-dimensional Euclidean space are combinatorially isomorphic (or combinatorially equivalent) if they have isomorphic posets of faces.

Counting the number $A(n,d)$ of equivalence classes of arrangements, given $n$ distinct hyperplanes in (affine) $\mathbb{R}^d,$ is definitely a hard problem. Note that the number of polytopes with $n$ facets is a subproblem. I'm just wondering whether some easy small cases of this have been worked out, either recently or historically.

Some related answers to this question are out there. For instance, http://oeis.org/A241600 counts the number of arrangements of lines in the affine plane. (Edit: Peter Shor points out below that this sequence is defined differently than via combinatorial equivalence.) Some nice pictures are included at http://oeis.org/A241600/a241600.pdf This gives a start to the question in dimension 3. The four arrangements of three lines ($A(3,2) =4$) can be extended along the third dimension to get arrangements of 3 planes in 3 dimensions, and then of course there is a fifth in which the three planes intersect at a point. These five ($A(3,3) = 5$) are pictured in many beginner algebra texts.

Drawing similar pictures, I think that $A(4,3) \ge 14.$ That is, that there are at least fourteen combinatorial equivalence classes of affine hyperplane arrangements using four planes in $\mathbb{R}^3.$ What else is known?