A260885 - OEIS

%I #23 Sep 08 2022 08:46:13

%S 1,2,1,6,6,2,21,62,37,0,97,559,788,112,0,579,5614,14558,7223,0,0,3812,

%T 56526,246331,277407,34748,0,0,27328,580860,3900740,8179658,3534594,0,

%U 0,0,206410,6020736,58842028,203974134,198559566,22524176,0,0,0

%N 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 oriented, the surface is unoriented).

%C 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.

%C Row g=0 is A008988 starting with n = 1.

%C 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

%H R. Coquereaux, J.-B. Zuber, <a href="http://arxiv.org/abs/1507.03163">Maps, immersions and permutations</a>, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474

%e The transposed triangle starts:

%e 1  2  6  21   97   579    3812    27328     206410

%e    1  6  62  559  5614   56526   580860    6020736

%e       2  37  788 14558  246331  3900740   58842028

%e           0  112  7223  277407  8179658  203974134

%e                0     0   34748  3534594  198559566

%e                      0       0        0   22524176

%e                              0        0          0

%e                                       0          0

%o (Magma) /* Example n := 6 */

%o n:=6;

%o n; // n: number of crossings

%o G:=Sym(2*n);

%o doubleG := Sym(4*n);

%o genH:={};

%o for j in [1..(n-1)] do v := G!(1,2*j+1)(2, 2*j+2); Include(~genH,v) ; end for;

%o H := PermutationGroup< 2*n |genH>; //  The H=S(n) subgroup of S(2n)

%o cardH:=#H;

%o cardH;

%o rho:=Identity(G); for j in [0..(n-1)] do v := G!(2*j+1, 2*j+2) ; rho := rho*v ; end for;

%o cycrho := PermutationGroup< 2*n |{rho}>; // The cyclic subgroup Z2 generated by rho (mirroring)

%o Hcycrho:=sub<G|[H,cycrho]>;  // The subgroup generated by H and cycrho

%o cardZp:= Factorial(2*n-1);

%o beta:=G!Append([2..2*n],1); // A typical circular permutation

%o Cbeta:=Centralizer(G,beta);

%o bool, rever := IsConjugate(G,beta,beta^(-1));

%o cycbeta := PermutationGroup< 2*n |{rever}>;

%o Cbetarev := sub<G|[Cbeta,cycbeta]>;

%o psifct := function(per);

%o perinv:=per^(-1);

%o res:= [IsOdd(j) select (j+1)^per  else j-1 + 2*n : j in [1..2*n] ];

%o 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] ];

%o res cat:= resbis;

%o return doubleG!res;

%o end function;

%o numberofcycles := function(per);   ess :=   CycleStructure(per); return &+[ess[i,2]: i in [1..#ess]]; end function;

%o supernumberofcycles := function(per); return  numberofcycles(psifct(per)) ; end function;

%o // result given as a list genuslist (n+2-2g)^^multiplicity where g is the genus

%o // Case OU

%o dbl, dblsize := DoubleCosetRepresentatives(G,Hcycrho,Cbeta); #dblsize;

%o genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist;

%o quit;

%o # _Robert Coquereaux_, Nov 23 2015

%Y Cf. A008988. The sum over all genera g for a fixed number n of crossings is given by sequence A260887. Cf. A260885, A260848, A260914.

%K nonn,tabl,hard

%O 1,2

%A _Robert Coquereaux_, Aug 02 2015