A111276 Number of chiral non-crossing partition patterns of n points on a circle, divided by 2.

0, 0, 0, 0, 0, 4, 14, 60, 210, 728, 2442, 8252, 27716, 93924, 319964, 1098900, 3800928, 13244836, 46460738, 164015272, 582353976, 2078812492, 7457141650, 26871707908, 97236327900, 353213328024, 1287648322950, 4709765510884, 17279999438748, 63583033400968

Offset: 1

Author

David Callan and Len Smiley, Oct 21 2005

Keywords

• nonn

Comments

Half of the number of those rotation-inequivalent patterns of non-crossing partitions of n (equally spaced) points on a circle which are not invariant under reflections. Division by two counts one pattern from each chiral (Right-handed,Left-handed) pair.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• D. Callan and L. Smiley, Non-crossing Partitions under Rotation and Reflection, arXiv:math/0510447 [math.CO], 2005.
• L. Smiley, a(5) = 0
• L. Smiley, a(6)=8/2=4

Crossrefs

Cf. A001405, A054357, A111275.

Programs

Mathematica

a[n_] := If[n < 6, 0, ((Binomial[2n, n]/(n+1) + DivisorSum[n, Binomial[2#, #] EulerPhi[n/#] Boole[# < n]&])/n - Binomial[n, Floor[n/2]])/2];
Array[a, 22] (* Jean-François Alcover, Feb 17 2019 *)

PARI

a(n) = (sumdiv(n, d, eulerphi(n/d)*binomial(2*d, d))/n - binomial(2*n, n)/(n+1) - binomial(n,n\2))/2 \\ Andrew Howroyd, Nov 19 2024

Formula

a(n) = (A054357(n) - A001405(n))/2.

Extensions

a(23) onwards from Andrew Howroyd, Nov 19 2024