[seqfan] Arranging circles on a sphere -- A000055?
Vladimir Reshetnikov
v.reshetnikov at gmail.com
Sat Aug 13 21:49:46 CEST 2016
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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