An idea+2 questions for A241600, the number of ways of arranging n lines in the (affine) plane.

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Hello all,

I've recently started looking at Neil Sloane's posed problem of A241600. I've been staring at these lines for the past week or so, and I have a suggestion for looking into these and a related question that I'd like to ask. First, I will explain the suggestion, then I will ask the questions far below.

Suggestion: focus only on the parallel lines, imagine their intersection point to be at infinity (I'm aware the problem is focused on the affine plane but bear with me), and imagine pulling that point to the affine plane. Pull it through an imaginary center line between the parallel lines, now non-parallel. If there are multiple lines, pull it through the centerline of all of them (I will address the issue of defining this centerline in a moment). As you pull this point, note the configurations before, at, and after a line/intersection point is crossed as a node in a graph. Repeat for all configurations of parallel lines.

By center line, there are 2 cases: For an even number of parallel lines, take it to be the line between the innermost 2, which is a problematic definition as seen below. For an odd number, take it to be the innermost line itself, which maaaay also be a problem (?).

When you make the graph, you end up with a bunch of sort of "cycles" that connects the various configurations to each other, with duplicate end points. There will be the end points which are configurations of parallel lines, and there will be pairs of edges that show these mirrored to each other. Below are illustrations of this idea.

For n = 0, 1, 2, the graphs are trivial.

For n = 2, you can see the basics of pulling the intersection at infinity to and from infinity.

At n = 3, you can see the example of what I'm trying to do here. Join up the top and bottom configurations since they are the exact same, and you get a pretty graph to start with. Notice the 3 and 1 counts the nodes outside the root node, which I consider to be the configuration where all the lines intersect at 1 point. I use this notation beyond to help keep track and count the nodes in these graphs.

At a(4), I thought I was starting to get an elegant pattern which could help with this problem. There's some complex structure you could start to see emerge from all the possible configurations, all of whom were accounted for! The 1, 3s, and 5s all count the possible configurations beyond the "root node" so to speak.

And for a(5), I did a part of the full set, since I noticed a pattern that led me to believe I could cut down on all the drawing. Again, the result is rather interesting!

I'm now looking at a(5) = 791. If the idea held merit, then my plan was to draw out all the possible parallel lines and begin the tedious work of making what appeared to be an elegant graph. However...

Where there are multiple pairs of parallel lines, this idea of a centerline breaks down completely. It doesn't account for where the centerline should cross. I don't know how this could break down for odd numbers of parallel lines, but I imagine it could happen the same way, just with the outer lines misbehaving.

Question 1:
This is where I throw this conundrum to the community. I have either 2 options, both of which are bad: redefine the definition of centerline to cross through any and all intersections, which doesn't account for the new centerline not being parallel with the lines themselves, or continue with this original definition and account for aaalll the possible ways "pulling the intersection at infinity" could occur, which would make this endeavor possibly grindy and delicate (and therefore probably incorrect?). What do you all think?

Question 2 (assuming Question 1 is answered in the positive):

Does anyone know or have posed a related problem for how many ways there are to arrange n lines such that all lines are parallel with at least 1 other line? If so, it would help greatly in figuring out what I'm dealing with for n >= 6.

Thank you all!
Nour Abouyoussef