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

1,2

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

Table of n, a(n) for n=1..28.

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)

Formula

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

Mathematica

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

Crossrefs

Cf. A209612.

Sequence in context: A185164 A124345 A123256 * A298537 A272079 A212606

Adjacent sequences: A111272 A111273 A111274 * A111276 A111277 A111278

Keyword

nonn

Author

David Callan and Len Smiley, Oct 21 2005

Status

approved