[seqfan] Re: Arranging circles on a sphere -- A000055?
Андрей Заболотский
zabolotis at mail.ru
Thu Aug 25 13:56:48 CEST 2016
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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