A260847 -id:A260847 - cp's OEIS frontend

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

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

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.

			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 *)

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

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

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

A260848 Triangle read by rows: T(n,g) = number of general immersions of a circle with n crossings in a surface of arbitrary genus g (the circle is not oriented, the surface is oriented).

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 21, 64, 36, 0, 99, 559, 772, 108, 0, 588, 5656, 14544, 7222, 0, 0, 3829, 56528, 246092, 277114, 34680, 0, 0, 27404, 581511, 3900698, 8180123, 3534038, 0, 0, 0, 206543, 6020787, 58838383, 203964446, 198551464, 22521600, 0, 0, 0

Offset: 1

Robert Coquereaux, Aug 01 2015

When transposed, displayed as an upper right triangle, the first line g = 0 of the table gives the number of immersions of a circle with n double points in a sphere (spherical curves) starting with n=1, the second line g = 1 gives immersions in a torus, etc.
Row g=0 is A008987 starting with n = 1.
For g > 0 the immersions are understood up to stable geotopy equivalence (the counted curves cannot be immersed in a surface of smaller genus). - Robert Coquereaux, Nov 23 2015

			The transposed triangle starts:
1    2   6   21    99     588      3829      27404            206543
     1   6   64   559    5656     56528     581511           6020787
         1   36   772   14544    246092    3900698          58838383
              0   108    7222    277114    8180123         203964446
                    0      0      34680    3534038         198551464
                           0          0         0           22521600
                                      0         0                  0
                                                0                  0

R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv:1507.03163 [math.CO], 2015, Table 9.

The sum over all genera g for a fixed number n of crossings is given by sequence A260847.

Cf. A008987, A260285, A260885, A260914.

/* Example n := 6 */
n:=6;
n; // n: number of crossings
G:=Sym(2*n);
doubleG := Sym(4*n);
genH:={};
for j in [1..(n-1)] do v := G!(1,2*j+1)(2, 2*j+2); Include(~genH,v) ; end for;
H := PermutationGroup< 2*n |genH>; //  The H=S(n) subgroup of S(2n)
cardH:=#H;
cardH;
rho:=Identity(G); for j in [0..(n-1)] do v := G!(2*j+1, 2*j+2) ; rho := rho*v ; end for;
cycrho := PermutationGroup< 2*n |{rho}>; // The cyclic subgroup Z2 generated by rho (mirroring)
Hcycrho:=sub;  // The subgroup generated by H and cycrho
cardZp:= Factorial(2*n-1);
beta:=G!Append([2..2*n],1); // A typical circular permutation
Cbeta:=Centralizer(G,beta);
bool, rever := IsConjugate(G,beta,beta^(-1));
cycbeta := PermutationGroup< 2*n |{rever}>;
Cbetarev := sub;
psifct := function(per);
perinv:=per^(-1);
res:= [IsOdd(j) select (j+1)^per  else j-1 + 2*n : j in [1..2*n] ];
resbis := [IsOdd((j-2*n)^perinv) select  (j-2*n)^perinv +1 +2*n   else ((j-2*n)^perinv -1)^per : j in [2*n+1..4*n] ];
res cat:= resbis;
return doubleG!res;
end function;
numberofcycles := function(per);   ess :=   CycleStructure(per); return &+[ess[i,2]: i in [1..#ess]]; end function;
supernumberofcycles := function(per); return  numberofcycles(psifct(per)) ; end function;
// result given as a list genuslist (n+2-2g)^^multiplicity where g is the genus
// Case UO
dbl, dblsize := DoubleCosetRepresentatives(G,H,Cbetarev); #dblsize;
genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist;
quit;
// Robert Coquereaux, Nov 23 2015

A260296 Sum over the genera g of the number of immersions of an oriented circle with n crossing in an oriented surface of genus g.

Original entry on oeis.org

1, 4, 22, 218, 3028, 55540, 1235526, 32434108, 980179566, 33522177088, 1279935820810, 53970628896500, 2490952020480012, 124903451391713412, 6761440164391403896, 393008709559373134184, 24412776311194951680016, 1613955767240361647220648, 113146793787569865523200018, 8384177419658944198600637096

Offset: 1

Robert Coquereaux, Jul 22 2015

nonn

a(n) is the sum over the n-th row of triangle A260285.
a(n) is also the number of double cosets of H\G/K where G is the symmetric group S(2n), H is the centralizer of a circular permutation of G, and K is a subgroup of G isomorphic with S(n) that commutes with
(1,2)(3,4)...(2n-3,2n-2)(2n-1,2n), using cycle notation, and permutes odd resp. even integers among themselves.
For n a prime integer, there is an explicit formula: a(n) = n-1 +(2n-1)!/n!.
For given g > 0 the immersions are understood up to stable geotopy equivalence (listed curves cannot be immersed in a surface of smaller genus). - Robert Coquereaux, Nov 23 2015

R. Coquereaux, J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI
R. J. Mathar, Chord Diagrams with Directed Chords, viXra:2410.0145 (2024)

Cf. A260285, A260912, A260847, A260887.

/* For n a prime integer */ [NthPrime(n)-1 +Factorial(2*NthPrime(n)-1) div Factorial(NthPrime(n)): n in [0..10]]; // Vincenzo Librandi, Aug 01 2015

/* For all n */
nbofdblecos := function(G, H, K);
CG := Classes(G); nCG := #CG; oG := #G; CH := Classes(H); nCH := #CH; oH := #H; CK := Classes(K); nCK := #CK; oK := #K;
resH := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Hmupositions := {j: j in [1..nCH]  | CycleStructure(CH[j][3]) eq CycleStructure(Gmurep)};
Hmugoodpositions := {j : j in Hmupositions | IsConjugate(G,CH[j][3], Gmurep) eq true}; bide := 0; for j in Hmugoodpositions do bide := bide + CH[j][2]; end for; Append(~resH, bide); end for;
resK := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Kmupositions := {j: j in [1..nCK]  | CycleStructure(CK[j][3]) eq CycleStructure(Gmurep)};
Kmugoodpositions := {j : j in Kmupositions | IsConjugate(G,CK[j][3], Gmurep) eq true}; bide := 0; for j in Kmugoodpositions do bide := bide + CK[j][2]; end for;  Append(~resK, bide); end for;
ndcl := 0; tot := 0; for mu in [1..nCG] do tot := tot + resH[mu]* resK[mu]/CG[mu][2]; end for;  ndcl:= tot *  oG/(oH * oK); return ndcl;
end function;
OOfull := function(n); G:=Sym(2*n); genH:={}; for j in [1..(n-1)] do v := G!(1,2*j+1)(2, 2*j+2); Include(~genH,v) ; end for;
H := PermutationGroup< 2*n |genH>;
beta:=G!Append([2..2*n],1); Cbeta:=Centralizer(G,beta);
return nbofdblecos(G, H, Cbeta); end function;
[OOfull(n) : n in [1..10]];
// Robert Coquereaux, Aug 01 2015

A260887 Sum over the genera g of the number of immersions of an oriented circle with n crossings in an unoriented surface of genus g.

Original entry on oeis.org

1, 3, 14, 120, 1556, 27974, 618824, 16223180, 490127050, 16761331644, 639969571892, 26985326408240, 1245476099801252, 62451726395242858, 3380720087847928728, 196504354827002278248, 12206388156005725243280, 806977883623811932432386, 56573396893818112613554940, 4192088709829783508863131872

Offset: 1

Robert Coquereaux, Aug 02 2015

nonn

a(n) is the sum over the n-th row of the triangle A260885.
a(n) is also the number of double cosets of H\G/K where G is the symmetric group S(2n), H is the subgroup generated by the centralizer of the circular permutation β = (1,2,3,...,2n) of G, K is a subgroup of G generated by the permutation ρ = (1,2)(3,4)...(2n-3,2n-2)(2n-1,2n), using cycle notation, and the subgroup (isomorphic with S(n)) that commutes with ρ and permutes odd resp. even integers among themselves.
For g > 0 the immersions are understood up to stable geotopy equivalence (the counted curves cannot be immersed in a surface of smaller genus). - Robert Coquereaux, Nov 23 2015

R. Coquereaux, J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474

Cf. A260296, A260847, A260885, A260912.

/* For all n */
nbofdblecos := function(G, H, K);
CG := Classes(G); nCG := #CG; oG := #G; CH := Classes(H); nCH := #CH; oH := #H; CK := Classes(K); nCK := #CK; oK := #K;
resH := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Hmupositions := {j: j in [1..nCH]  | CycleStructure(CH[j][3]) eq CycleStructure(Gmurep)};
Hmugoodpositions := {j : j in Hmupositions | IsConjugate(G,CH[j][3], Gmurep) eq true}; bide := 0; for j in Hmugoodpositions do bide := bide + CH[j][2]; end for;
Append(~resH, bide); end for;
resK := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Kmupositions := {j: j in [1..nCK]  | CycleStructure(CK[j][3]) eq CycleStructure(Gmurep)};
Kmugoodpositions := {j : j in Kmupositions | IsConjugate(G,CK[j][3], Gmurep) eq true}; bide := 0; for j in Kmugoodpositions do bide := bide + CK[j][2]; end for;
Append(~resK, bide); end for;
ndcl := 0; tot := 0; for mu in [1..nCG] do tot := tot + resH[mu]* resK[mu]/CG[mu][2]; end for;  ndcl:= tot *  oG/(oH * oK); return ndcl;
end function;
OUfull := function(n); G:=Sym(2*n); genH:={}; for j in [1..(n-1)] do v := G!(1,2*j+1)(2, 2*j+2); Include(~genH,v) ; end for;
H := PermutationGroup< 2*n |genH>;
beta:=G!Append([2..2*n],1); Cbeta:=Centralizer(G,beta);
rho:=Identity(G);  for j in [0..(n-1)] do v := G ! (2*j+1, 2*j+2) ; rho := rho*v ; end for;
cycrho := PermutationGroup< 2*n |{rho}>;  Hcycrho:=sub;
return nbofdblecos(G,Hcycrho,Cbeta); end function;
[OUfull(n) : n in [1..10]]; //

A260912 Sum over the genera g of the number of immersions of an unoriented circle with n crossings in an unoriented surface of genus g.

Original entry on oeis.org

1, 3, 12, 86, 894, 14715, 313364, 8139398, 245237925, 8382002270, 319994166042, 13492740284184, 622738642693202, 31225868370080949, 1690360086869176780, 98252177808632109236, 6103194081506193327048, 403488941845715112039425, 28286698447226523233226110, 2096044354918091666701275248

Offset: 1

Robert Coquereaux, Aug 04 2015

nonn

a(n) is the sum over the n-th row of triangle A260914.
a(n) is also the number of double cosets of H\G/K where G is the symmetric group S(2n), H is the subgroup generated by the centralizer of the circular permutation β = (1,2,3,...,2n) of G together with the permutation (2, 2n)(3, 2n-1)(4, 2n-2)...(n, n+2) that conjugates β and β-1, K is a subgroup of G generated by the permutation ρ = (1,2)(3,4)...(2n-3,2n-2)(2n-1,2n), using cycle notation, and the subgroup (isomorphic with S(n)) that commutes with ρ and permutes odd resp. even integers among themselves.
For g > 0 the immersions are understood up to stable geotopy equivalence (the counted curves cannot be immersed in a surface of smaller genus). - Robert Coquereaux, Nov 23 2015

R. Coquereaux, J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474

Cf. A260847, A260887, A260296, A260914.

/* For all n */
nbofdblecos := function(G, H, K);
CG := Classes(G); nCG := #CG; oG := #G; CH := Classes(H); nCH := #CH; oH := #H; CK := Classes(K); nCK := #CK; oK := #K;
resH := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Hmupositions := {j: j in [1..nCH]  | CycleStructure(CH[j][3]) eq CycleStructure(Gmurep)};
Hmugoodpositions := {j : j in Hmupositions | IsConjugate(G,CH[j][3], Gmurep) eq true}; bide := 0; for j in Hmugoodpositions do bide := bide + CH[j][2]; end for; Append(~resH, bide); end for;
resK := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Kmupositions := {j: j in [1..nCK]  | CycleStructure(CK[j][3]) eq CycleStructure(Gmurep)};
Kmugoodpositions := {j : j in Kmupositions | IsConjugate(G,CK[j][3], Gmurep) eq true}; bide := 0; for j in Kmugoodpositions do bide := bide + CK[j][2]; end for; Append(~resK, bide); end for;
ndcl := 0; tot := 0; for mu in [1..nCG] do tot := tot + resH[mu]* resK[mu]/CG[mu][2]; end for;  ndcl:= tot *  oG/(oH * oK); return ndcl;
end function;
UUfull := function(n); G:=Sym(2*n); genH:={}; for j in [1..(n-1)] do v := G!(1,2*j+1)(2, 2*j+2); Include(~genH,v) ; end for;
H := PermutationGroup< 2*n |genH>;
beta:=G!Append([2..2*n],1); Cbeta:=Centralizer(G,beta); bool, rever := IsConjugate(G,beta,beta^(-1)); cycbeta := PermutationGroup< 2*n |{rever}>;
Cbetarev := sub;
rho:=Identity(G);  for j in [0..(n-1)] do v := G ! (2*j+1, 2*j+2) ; rho := rho*v ; end for; cycrho := PermutationGroup< 2*n |{rho}>;
Hcycrho:=sub;
return nbofdblecos(G,Hcycrho,Cbetarev); end function;
[UUfull(n) : n in [1..10]]; //

cp's OEIS Frontend

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

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A260848 Triangle read by rows: T(n,g) = number of general immersions of a circle with n crossings in a surface of arbitrary genus g (the circle is not oriented, the surface is oriented).

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Magma

A260296 Sum over the genera g of the number of immersions of an oriented circle with n crossing in an oriented surface of genus g.

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A260887 Sum over the genera g of the number of immersions of an oriented circle with n crossings in an unoriented surface of genus g.

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A260912 Sum over the genera g of the number of immersions of an unoriented circle with n crossings in an unoriented surface of genus g.

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