A054499 Number of pairings on a bracelet; number of chord diagrams that can be turned over and having n chords.

1, 1, 2, 5, 17, 79, 554, 5283, 65346, 966156, 16411700, 312700297, 6589356711, 152041845075, 3811786161002, 103171594789775, 2998419746654530, 93127358763431113, 3078376375601255821, 107905191542909828013, 3997887336845307589431

Offset: 0

Author

Christian G. Bower, Apr 06 2000 based on a problem by Wouter Meeussen

Keywords

• nonn
• easy
• nice

Comments

Place 2n points equally spaced on a circle. Draw lines to pair up all the points so that each point has exactly one partner. Allow turning over.

Examples

			For n=3, there are 5 bracelets with 3 pairs of beads. They are represented by the words aabbcc, aabcbc, aabccb, abacbc, and abcabc. All of the 6!/(2*2*2) = 90 combinations can be derived from these by some combination of relabeling the pairs, rotation, and reflection. So a(3) = 5. - _Michael B. Porter_, Jul 27 2016
		

References

• R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

Links

• W. Y.-C. Chen, D. C. Torney, Equivalence classes of matchings and lattice-square designs, Discr. Appl. Math. 145 (3) (2005) 349-357.
• Étienne Ghys, A Singular Mathematical Promenade, arXiv:1612.06373 [math.GT], 2016-2017. See p. 252.
• A. Khruzin, Enumeration of chord diagrams, arXiv:math/0008209 [math.CO], 2000.
• V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.
• R. J. Mathar, Chord Diagrams A054499 (2018)
• R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019)
• R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)
• Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233.
• Index entries for sequences related to bracelets

Crossrefs

Cf. A007769, A104256, A279207, A279208, A003437 (loopless chord diagrams), A322176 (marked chords), A362657, A362658, A362659 (three, four, five instances of each color rather than two), A371305 (Multiset Transf.), A260847 (directed chords).

Programs

Mathematica

max = 19;
alpha[p_, q_?EvenQ] := Sum[Binomial[p, 2*k]*q^k*(2*k-1)!!, {k, 0, max}];
alpha[p_, q_?OddQ] := q^(p/2)*(p-1)!!;
a[0] = 1;
a[n_] := 1/4*(Abs[HermiteH[n-1, I/2]] + Abs[HermiteH[n, I/2]] + (2*Sum[Block[{q = (2*n)/p}, alpha[p, q]*EulerPhi[q]], {p, Divisors[ 2*n]}])/(2*n));
Table[a[n], {n, 0, max}] (* Jean-François Alcover, Sep 05 2013, after R. J. Mathar; corrected by Andrey Zabolotskiy, Jul 27 2016 *)

Formula

a(n) = (2*A007769(n) + A047974(n) + A047974(n-1))/4 for n > 0.

Extensions

Corrected and extended by N. J. A. Sloane, Oct 29 2006

a(0)=1 prepended back again by Andrey Zabolotskiy, Jul 27 2016