[seqfan] Re: Arranging circles on a sphere -- A000055?

[Jon Wild]

On Thu, 25 Aug 2016, Richard J. Mathar wrote:

> For circles in the flat plane topology I have used that sort of counting
> extensively in http://arxiv.org/abs/1603.00077 .

Richard, this is very nice!

You might be interested to know that in the general case (i.e. A250001), 
once you get to n=5, geometric concerns mean that not all 
topologically-conceivable arrangements are actually circle-drawable.

My program enumerated 16968 conceivable arrangements of 5 pseudo-circles, 
and Christopher Jones and I together have figured out how to show that 26 
of these arrangements are not actually circle-drawable. We haven't written 
anything up yet but I hope to get to it soon.

The spherical aficionados among you might like to know that the connected 
5-arrangements (ones where you can reach any circle from any other circle 
via a chain of intersections, which number 15528 among the 16968 planar 
5-arrangements) collapse into 984 arrangements when reprojected onto the 
surface of a sphere. Of these 984 spherical arrangements, all but 4 are 
circle-drawable. (The four undrawable spherical arrangements reproject 
into the 26 different undrawable planar arrangements.)

Regards,

Jon Wild