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A186952 Number of partitions of n concentric circles on the 2-sphere which are realizable by surfaces in the 3-ball 0
1, 1, 2, 4, 9, 20, 48, 113, 282, 689, 1767, 4435, 11616, 29775, 79352, 206960, 559906 (list; graph; refs; listen; history; text; internal format)



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.


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


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.


Sequence in context: A199883 A036624 A226907 * A034823 A036625 A003019

Adjacent sequences:  A186949 A186950 A186951 * A186953 A186954 A186955




Kevin Walker, Mar 01 2011



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Last modified November 13 23:01 EST 2019. Contains 329106 sequences. (Running on oeis4.)