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

Vladimir Reshetnikov v.reshetnikov at gmail.com
Thu Aug 25 20:21:36 CEST 2016

Yes, In this problem I deliberately consider a sphere without any special
points -- there are no poles or equator, and there is no intrinsic
difference between inside or outside region of any circle.


On Thu, Aug 25, 2016 at 4:56 AM, Андрей Заболотский via SeqFan <
seqfan at list.seqfan.eu> wrote:

>  It is true, but it does not contradict what Vladimir says. When circles
> are placed on a plane, insidedness is indeed a natural thing. But sphere
> has no special point and no special outside region, so it is natural to
> deny the insidedness in this case. And that is why the trees with unlabeled
> nodes instead of the rooted ones appear here.
> Equivalently, we may consider a plane but allow inversion symmetry
> operations. Inversions preserve the tree but changes its root, if I am not
> mistaken.
> (Perhaps it was clear for you, and I just miss something in your
> reasoning; I'm sorry if it is the case.)
> Четверг, 25 августа 2016, 13:18 +03:00 от "Richard J. Mathar" <
> mathar at mpia-hd.mpg.de >:
> >In  http://list.seqfan.eu/pipermail/seqfan/2016-August/016630.html
> Vladimir spake:
> >
> >vr> What is the number of ways to arrange n unlabeled non-intersecting
> circles
> >vr> on a sphere? Two arrangements are considered equivalent iff they can
> be
> >vr> transformed to one another by a combination of the following motions:
> >vr> (1) reflection, (2) continuously moving circles, (3) continuously
> changing
> >vr> their radii, provided that the circles always stay non-intersecting
> and lie
> >vr> on the sphere.
> >
> >This entirely demolishes the helpful principle of insidedness:
> >Imagine two circles on the North polar cap with two slightly different
> radii.
> >Looking from the North on it the smaller circles is inside the larger.
> >Them move both differentially with intermediate widening (keeping the
> common axis)
> >over the equator to the South pole. Looking from the South on it the
> >circle that was inside the other is now outside the other. So the count
> >for two circles is 1 (?)
> >
> >vr> After some thought, it occurred to me that there is an isomorphism
> between
> >vr> an arrangement of circles and a tree with unlabeled nodes ��� the
> fragments
> >vr> of the sphere separated by circles correspond to the nodes of the
> tree, and
> >vr> ...
> >
> >For circles in the flat plane topology I have used that sort of counting
> >extensively in  http://arxiv.org/abs/1603.00077 . If "insidedness"
> >is not a "good quantum number," those rules do not apply any longer and
> the
> >principle of strong ordering (in trees, based on insidedness) is not
> useful
> >and must be replaced by something else.
> >
> >--
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