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

[Андрей Заболотский]

For me, this seems to be right. It also seems that the reflections don't matter here. (See also  https://oeis.org/A000081  for the case of a plane.)

Probably exactly the same is
 true for non-intersecting arrangements of D-dimensional spheres on a D+1-dimensional sphere for any D>0. It is reasonable to consider a D=0 analogue which is  https://oeis.org/A054357  (with the interpretation given in the comment), if I'm not mistaken.


BR
Andrey Zabolotskiy

>Суббота, 13 августа 2016, 22:49 +03:00 от Vladimir Reshetnikov < v.reshetnikov at gmail.com >:
>
>Dear SeqFans,
>
>
>
>I was thinking on the following problem:
>
>
>
>What is the number of ways to arrange n unlabeled non-intersecting circles
>on a sphere? Two arrangements are considered equivalent iff they can be
>transformed to one another by a combination of the following motions:
>(1) reflection, (2) continuously moving circles, (3) continuously changing
>their radii, provided that the circles always stay non-intersecting and lie
>on the sphere.
>
>
>
>After some thought, it occurred to me that there is an isomorphism between
>an arrangement of circles and a tree with unlabeled nodes — the fragments
>of the sphere separated by circles correspond to the nodes of the tree, and
>the circles correspond to the edges. An edge connects two nodes, iff the
>corresponding circle is the common boundary of two fragments.
>Simply-connected fragments ("caps") correspond to the leaf nodes.
>
>
>
>So it seems the sequence I'm looking for is just  http://oeis.org/A000055 .
>Am I right? Do we get the same result if we exclude reflection from the
>allowed motions?
>
>
>
>--
>
>Thanks
>
>Vladimir Reshetnikov
>
>--
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