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

[Richard J. Mathar]

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.