A186952 Number of partitions of n concentric circles on the 2-sphere which are realizable by surfaces in the 3-ball

1, 1, 2, 4, 9, 20, 48, 113, 282, 689, 1767, 4435, 11616, 29775, 79352, 206960, 559906

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.

Links

Table of n, a(n) for n=0..16.

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

Sequence in context: A199883 A036624 A226907 * A034823 A036625 A003019

Adjacent sequences: A186949 A186950 A186951 * A186953 A186954 A186955

Keyword

nonn

Author

Kevin Walker, Mar 01 2011

Status

approved