A111275 Number of inequivalent non-crossing partitions of n (equally spaced) points on a circle, under rotations and reflections.

1, 2, 3, 6, 10, 24, 49, 130, 336, 980, 2904, 9176, 29432, 97356, 326399, 1111770, 3825238, 13293456, 46553116, 164200028, 582706692, 2079517924, 7458493728, 26874412064, 97241528200, 353223728624, 1287668381250, 4709805627484

Offset: 1

Author

David Callan and Len Smiley, Oct 21 2005

Keywords

• nonn

Comments

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.).

References

• 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]
• 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.

Links

• D. Callan and L. Smiley, Non-crossing Partitions under Rotation and Reflection
• Tilman Piesk, Partition related number triangles
• L. Smiley, a(6)
• L. Smiley, a(5)

Crossrefs

Cf. A209612.

Programs

Mathematica

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

Formula

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