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Number of inequivalent non-crossing partitions of n (equally spaced) points on a circle, under rotations and reflections.
8

%I #17 Oct 31 2015 14:01:38

%S 1,2,3,6,10,24,49,130,336,980,2904,9176,29432,97356,326399,1111770,

%T 3825238,13293456,46553116,164200028,582706692,2079517924,7458493728,

%U 26874412064,97241528200,353223728624,1287668381250,4709805627484

%N Number of inequivalent non-crossing partitions of n (equally spaced) points on a circle, under rotations and reflections.

%C These may be viewed as bracelets (able to be turned over in space) designed with n beads on a circle, each of which is a vertex of exactly one of a set of non-touching internal polygons (which may be 1-gons (beads), 2-gons (2 connected beads), etc.).

%D S.-C. Chang, J. L. Jacobsen, J. Salas, R. Shrock, "Exact Potts model partition functions for strips of the triangular lattice", J. Statist. Phys. 114, nos.3-4, pp. 763-823 [Corollary 2.1]

%D Motzkin, T. "Relations Between Hypersurface Cross Ratios and a Combinatorial Formula for Partitions of a Polygon for Permanent Preponderance and for Non-Associative Products." Bull. Amer. Math. Soc. 54, page 360, 1948.

%H D. Callan and L. Smiley, <a href="http://arXiv.org/abs/math.CO/0510447">Non-crossing Partitions under Rotation and Reflection</a>

%H Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Partition_related_number_triangles#rotref">Partition related number triangles</a>

%H L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/nc/6pointNCPP.pdf">a(6)</a>

%H L. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/nc/10NC.pdf">a(5)</a>

%F (A054357(n) + A001405(n))/2.

%t Table[Length[EquivalenceClasses[NCPartitions[n], groupDihedral[n]]], {n, 9}]

%Y Cf. A209612.

%K nonn

%O 1,2

%A _David Callan_ and _Len Smiley_, Oct 21 2005