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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?

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    $\begingroup$ 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 $\endgroup$
    – Zach Teitler
    Commented Aug 9, 2018 at 4:46
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    $\begingroup$ 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 $\endgroup$
    – Sam Hopkins
    Commented Aug 28, 2018 at 21:49
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    $\begingroup$ 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 $n$. $\endgroup$
    – Peter Shor
    Commented Aug 28, 2018 at 22:43
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    $\begingroup$ This presentation has a suggestive title: "Counting Hyperplane Arrangements," T. Paul, W. Traves, M. Wakefield. 2012. PDF download. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 28, 2018 at 22:56
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    $\begingroup$ 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. $\endgroup$
    – Peter Shor
    Commented Aug 29, 2018 at 21:22

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Combinatorial types of hyperplane arrangements are called dissection types in the dissertation of L. Finschi: https://finschi.com/math/publ/2001-08-31_Finschi_A-Graph-Theoretical-Approach-for-Reconstruction-and-Generation-of-Oriented-Matroids.pdf

This thesis is a great source for the connection between oriented matroids and the known answers to this question. For instance the first few terms in sequence of numbers of arrangements of $n$ 2D planes in 3 dimensions, up to combinatorial equivalence, can be found from Table 8.1 on page 165. Sum the first three rows to get the sequence 1, 2, 5, 14, 74, 1854, ...

[edit] Note however that this table (probably for higher $n$) counts pseudo-hyperplane arrangements, so the question is still open how many of each are realizable as hyperplane arrangements.

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