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%I #15 Mar 31 2012 10:25:48
%S 1,1,2,4,9,20,48,113,282,689,1767,4435,11616,29775,79352,206960,559906
%N Number of partitions of n concentric circles on the 2-sphere which are realizable by surfaces in the 3-ball
%C This is a higher dimensional version of non-crossing partitions and Catalan numbers. Given an arrangement of n circles on the 2-sphere, we can consider an unoriented surface in the 3-ball whose boundary is the given circles. Given such a surface, we get a partition of the circles by saying that two circles are in the same block if they are part of the boundary of a single connected component of the surface. The possible circle arrangements (up to isomorphism) are in bijection with unrooted trees with n edges, so we have a function from unrooted trees to the positive integers. This sequence is for linear trees with n edges and maximum valence 2.
%e For n=3, the allowable partitions are ABC, AAB, ABB, and AAA. For n=4 the allowable partitions are ABCD, ABCC, ABBC, AABC, AABB, ABBA, ABBB, AAAB, and AAAA.
%K nonn
%O 0,3
%A _Kevin Walker_, Mar 01 2011
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