A054499
Number of pairings on a bracelet; number of chord diagrams that can be turned over and having n chords.
Original entry on oeis.org
1, 1, 2, 5, 17, 79, 554, 5283, 65346, 966156, 16411700, 312700297, 6589356711, 152041845075, 3811786161002, 103171594789775, 2998419746654530, 93127358763431113, 3078376375601255821, 107905191542909828013, 3997887336845307589431
Offset: 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
- R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
- 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
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).
-
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 *)
A007769
Number of chord diagrams with n chords; number of pairings on a necklace.
Original entry on oeis.org
1, 1, 2, 5, 18, 105, 902, 9749, 127072, 1915951, 32743182, 624999093, 13176573910, 304072048265, 7623505722158, 206342800616597, 5996837126024824, 186254702826289089, 6156752656678674792, 215810382466145354405, 7995774669504366055054
Offset: 0
Jean.Betrema(AT)labri.u-bordeaux.fr
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- D. Bar-Natan, On the Vassiliev Knot Invariants, Topology 34 (1995) 423-472.
- D. Bar-Natan, Bibliography of Vassiliev Invariants.
- W. Y.-C. Chen, D. C. Torney, Equivalence classes of matchings and lattice-square designs, Discr. Appl. Math. 145 (3) (2005) 349-357., table of C_2n.
- Combinatorial Object Server, Information on Chord Diagrams
- Étienne Ghys, A Singular Mathematical Promenade, arXiv:1612.06373 [math.GT], 2016. See p. 252.
- A. Khruzin, Enumeration of chord diagrams, arXiv:math/0008209 [math.CO], 2000.
- R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section V.
- Joe Sawada, A fast algorithm for generating nonisomorphic chord diagrams, SIAM J. Discrete Math, Vol. 15, No. 4, 2002, pp. 546-561.
- Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233.
- Index entries for sequences related to necklaces
-
alpha:=proc(p, q)
local k;
if is(q, even) then
add(binomial(p, 2*k)*q^k*doublefactorial(2*k-1), k=0..p/2)
else
q^(p/2)*doublefactorial(p-1)
end if
end proc:
A007769 := proc(n)
local p;
if n = 0 then
1;
else
add(alpha(2*n/p, p)*numtheory[phi](p), p=numtheory[divisors](2*n))/2/n
end if;
end proc:
seq(A007769(n),n=0..10) ; # Robert FERREOL, Oct 10 2018
-
max = 20; alpha[p_, q_?EvenQ] := Sum[Binomial[p, 2k]*q^k*(2k-1)!!, {k, 0, max}]; alpha[p_, q_?OddQ] := q^(p/2)*(p-1)!!; a[0] = 1; a[n_] := Sum[q = 2n/p; alpha[p, q]*EulerPhi[q], {p, Divisors[2n]}]/(2n); Table[a[n], {n, 0, max}] (* Jean-François Alcover, May 07 2012, after R. J. Mathar *)
Stoimenow states that a Mma package is available from his website. - N. J. A. Sloane, Jul 26 2018
-
doublefactorial(n)={ local(resul) ; resul=1 ; forstep(i=n,2,-2, resul *= i ;) ; return(resul) ; }
alpha(n,q)={ if(q %2, return( q^(p/2)*doublefactorial(p-1)), return( sum(k=0,p/2,binomial(p,2*k)*q^k*doublefactorial(2*k-1)) ) ;) ; }
A007769(n)={ local(resul,q) ; if(n==0, return(1), resul=0 ; fordiv(2*n,p, q=2*n/p ; resul += alpha(p,q)*eulerphi(q) ;); return(resul/(2*n)) ;) ; } { for(n=0,20, print(n," ",A007769(n)) ;) ; } \\ R. J. Mathar, Oct 26 2006
A279207
Number of analytic chord diagrams with n chords.
Original entry on oeis.org
1, 2, 5, 18, 102, 817, 7641
Offset: 1
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