[seqfan] Re: Number of arrangements of n lines in the plane, A241600

[Jon Wild]

Dear Neil,

I tried computing a very similar sequence at some point--it's different 
from yours in that I wasn't interested (at that time) in arrangements that 
included more than two lines crossing in a single point, and I required 
the lines to cross every other line. I submitted it (with Laurence Reeves) 
as A090338 in 2004. I'm confident of the results up to 1, 1, 1, 1, 1, 6, 
43, 922, but I am no longer sure if the next term (8 lines) should have 
been 38609 or 38612. At some point you can have an abstract arrangement 
that can't be realised with straight lines, and we thought we could show 
that that was the case for 3 of the n=8 arrangements. In A090339 we threw 
out the requirement that the lines be straight; one of the three 
arrangements that we thought was only possible with curved lines is shown 
here:

https://oeis.org/A090339/a090339.gif

Afterwards I remember something made me unsure of that proof, but it's a 
little hazy in my mind and I'm no longer in touch with Laurence Reeves. I 
can't remember what we did for handedness! (I mean, I don't remember if 
reflections were counted as equivalent). I also can't remember the 
derivation of the name we gave these arrangements--"flups". (An acronym 
for "flat, undirected p[something]", I think.)

We described these arrangements as collections of line *segments*, that 
all crossed one another (with no triple points and no "T" junctions), 
rather than arrangements of (infinite) lines. The next sequence would have 
been those arrangements where lines do not have to all cross one another, 
which I discovered a neat method for enumerating, once you have all the 
fully crossing arrangements (which are simple to count if you don't care 
about the straight-line condition)--but we never quite finished generating 
the results cleanly before my collaborator disappeared. I have an older 
version done by hand up to n=5, which if I remember correctly had about 
270-280 possibilities.

I also did some work on the number of arrangements of circles in the 
plane, under various conditions, which I need to turn into a proper entry 
at some point. Here's a picture of the first 60 or so arrangements for 
n=4:

http://www.music.mcgill.ca/~wild/circles_nEquals4_page1.png

When I first counted these arrangements I was very tickled to see the 
sequence started (for n=2, 3, 4...) as 3,14,159. If the next entry had 
been 2653 I would have had a heart attack (decimal expansion of pi). But 
on recomputing for n=4 I found I had missed a few, and the true result is 
168, not 159.

Jon Wild



On Thu, 15 May 2014, Neil Sloane wrote:

> Dear Seq Fans,
> Max Alekseyev and I were looking at this question.
> I computed the first five terms, by hand, so they are
> probably wrong. To get the ball rolling, I created A241600.
> Comments and corrections are welcomed!
> Neil
>