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

[Neil Sloane]

Jon, Thanks for that reply! I was happy to see that you also found 6
arrangements of 5 lines in general position.

I added your "circles" version as A241601. You don't allow
circles to meet in one point, or a point to be on more than 2 circles.

As I commented in A241601, there are several other cases that could be
considered:

Several variations are possible:

- straight lines instead of circles (see A241600 <https://oeis.org/A241600>
).

- straight lines in general position (A090338 <https://oeis.org/A090338>).

- curved lines in general position (A090339 <https://oeis.org/A090339>).

- allow circles to meet tangentially but without multiple intersection
points (begins 1, 5, ...)

- again use circles, but allow multiple intersection points (also begins 1,
5, ...).

- use ellipses rather than circles.

- allow lines OR circles (since projectively there is no difference between
a line and a circle).


Lots of potentially new sequences here for anyone who wants to play with
this problem - and the existing ones all need more terms.

And programs - none of them have programs!


Neil








On Thu, May 15, 2014 at 10:08 PM, Jon Wild <wild at music.mcgill.ca> wrote:

>
> 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
>>
>>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>



-- 
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com