# Counting hyperplane arrangements up to combinatorial equivalence, simple examples and history

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?

• Maybe related, this work on the number of matroids of a given rank on a ground set of a given cardinality: homepages.dias.ie/dukes/matroid.html – Zach Teitler Aug 9 '18 at 4:46
• Not directly related to your question, but if you fix some (multi)set of linear hyperplanes and then look at all affine hyperplane arrangements obtained from this set by translating some hyperplanes along their normals, then this moduli space naturally has the structure of a hyperplanes arrangement, which is called (if I recall correctly) the “discriminantal arrangement” of your set of linear hyperplanes – Sam Hopkins Aug 28 '18 at 21:49
• A241600 does not count the number of arrangements that are equivalent according to your definition. It counts the number that are equivalent with the definition "Two arrangements are considered the same if one can be continuously changed to the other while keeping all lines straight, without changing the multiplicity of intersection points, and without a line passing through an intersection point. Turning over is also allowed." These two definitions are different for large enough $n$. – Peter Shor Aug 28 '18 at 22:43
• This presentation has a suggestive title: "Counting Hyperplane Arrangements," T. Paul, W. Traves, M. Wakefield. 2012. PDF download. – Joseph O'Rourke Aug 28 '18 at 22:56
• I have no idea whether anybody has figured out the smallest pair of such pseudo-line arrangements. So it's quite possible that the sequence A241600 coincides with the sequence you're interested in up to $n=7$ (the highest term in the OEIS). I'm tempted to ask the question of whether anybody knows a small such pair on MathOverflow. – Peter Shor Aug 29 '18 at 21:22