This site is supported by donations to The OEIS Foundation.



Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A054499 Number of pairings on a bracelet; number of chord diagrams that can be turned over and having n chords. 16
1, 1, 2, 5, 17, 79, 554, 5283, 65346, 966156, 16411700, 312700297, 6589356711, 152041845075, 3811786161002, 103171594789775, 2998419746654530, 93127358763431113, 3078376375601255821, 107905191542909828013, 3997887336845307589431 (list; graph; refs; listen; history; text; internal format)



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.


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


Table of n, a(n) for n=0..20.

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


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


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


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 *)


Cf. A007769, A104256, A279207, A279208, A003437 (loopless chord diagrams), A322176 (marked chords).

Sequence in context: A289739 A243337 A259622 * A001186 A125282 A020125

Adjacent sequences:  A054496 A054497 A054498 * A054500 A054501 A054502




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


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

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



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 10 23:29 EST 2019. Contains 329910 sequences. (Running on oeis4.)