cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A260914 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 not oriented).

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 19, 45, 22, 0, 76, 335, 427, 56, 0, 376, 3101, 7557, 3681, 0, 0, 2194, 29415, 124919, 139438, 17398, 0, 0, 14614, 295859, 1921246, 4098975, 1768704, 0, 0, 0, 106421, 3031458, 29479410, 102054037, 99304511, 11262088, 0, 0, 0
Offset: 1

Views

Author

Robert Coquereaux, Aug 04 2015

Keywords

Comments

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 A008989 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

Examples

			The transposed triangle starts:
  1  2  6  19   76   376     2194     14614     106421
     1  5  45  335  3101    29415    295859    3031458
        1  22  427  7557   124919   1961246   29479410
            0   56  3681   139438   4098975  102054037
                 0    0     17398   1768704   99394511
                      0         0         0   11262088
                                0         0          0
                                          0          0
                                                     0

Links

  • Robert Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: 10.1142/S0218216516500474

Crossrefs

The sum over all genera g for a fixed number n of crossings is given by sequence A260912. Cf. A008989, A260285, A260848, A260885.

Programs

  • Magma
    /* 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 UU
dbl, dblsize := DoubleCosetRepresentatives(G,Hcycrho,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

Views

Author

Robert Coquereaux, Jul 22 2015

Keywords

Comments

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

Links

Crossrefs

Cf. A260285, A260912, A260847, A260887.

Programs

  • Magma
    /* 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
  • Magma
    /* 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

A260847 Sum over the genera g of the number of immersions of an unoriented circle with n crossing in an oriented surface of genus g.

Original entry on oeis.org

1, 3, 13, 121, 1538, 28010, 618243, 16223774, 490103223, 16761330464, 639968394245, 26985325092730, 1245476031528966, 62451726249369666, 3380720083302727868, 196504354812897344692, 12206388155663897395208, 806977883622439156487124, 56573396893789449427353609, 4192088709829643732598955348
Offset: 1

Views

Author

Robert Coquereaux, Aug 01 2015

Keywords

Comments

a(n) is the sum over the n-th row of triangle A260848.
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, 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 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

Links

Crossrefs

Cf. A260848, A260296, A260912, A260887.

Programs

  • Magma
    /* 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;
UOfull := 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; return nbofdblecos(G,H,Cbetarev); end function;
[UOfull(n) : n in [1..10]]; //

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

Views

Author

Robert Coquereaux, Aug 02 2015

Keywords

Comments

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

Links

  • 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

Crossrefs

Cf. A260296, A260847, A260885, A260912.

Programs

  • Magma
    /* 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]]; //
Showing 1-4 of 4 results.

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