OFFSET
0,3
COMMENTS
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.
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Kevin Walker, Mar 01 2011
STATUS
approved