cp's OEIS Frontend

A260885 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).

%I A260885 #28 Jan 25 2025 13:08:47
%S A260885 1,2,1,6,6,2,21,62,37,0,97,559,788,112,0,579,5614,14558,7223,0,0,3812,
%T A260885 56526,246331,277407,34748,0,0,27328,580860,3900740,8179658,3534594,0,
%U A260885 0,0,206410,6020736,58842028,203974134,198559566,22524176,0,0,0
%N A260885 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 A260885 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 A260885 Row g=0 is A008988 starting with n = 1.
%C A260885 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 A260885 Robert Coquereaux and J.-B. Zuber, <a href="http://arxiv.org/abs/1507.03163">Maps, immersions and permutations</a>, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474
%e A260885 The transposed triangle starts:
%e A260885   1  2  6  21   97   579    3812    27328     206410
%e A260885      1  6  62  559  5614   56526   580860    6020736
%e A260885         2  37  788 14558  246331  3900740   58842028
%e A260885             0  112  7223  277407  8179658  203974134
%e A260885                  0     0   34748  3534594  198559566
%e A260885                        0       0        0   22524176
%e A260885                                0        0          0
%e A260885                                         0          0
%e A260885                                                    0
%o A260885 (Magma) /* Example n := 6 */
%o A260885 n:=6;
%o A260885 n; // n: number of crossings
%o A260885 G:=Sym(2*n);
%o A260885 doubleG := Sym(4*n);
%o A260885 genH:={};
%o A260885 for j in [1..(n-1)] do v := G!(1,2*j+1)(2, 2*j+2); Include(~genH,v) ; end for;
%o A260885 H := PermutationGroup< 2*n |genH>; //  The H=S(n) subgroup of S(2n)
%o A260885 cardH:=#H;
%o A260885 cardH;
%o A260885 rho:=Identity(G); for j in [0..(n-1)] do v := G!(2*j+1, 2*j+2) ; rho := rho*v ; end for;
%o A260885 cycrho := PermutationGroup< 2*n |{rho}>; // The cyclic subgroup Z2 generated by rho (mirroring)
%o A260885 Hcycrho:=sub<G|[H,cycrho]>;  // The subgroup generated by H and cycrho
%o A260885 cardZp:= Factorial(2*n-1);
%o A260885 beta:=G!Append([2..2*n],1); // A typical circular permutation
%o A260885 Cbeta:=Centralizer(G,beta);
%o A260885 bool, rever := IsConjugate(G,beta,beta^(-1));
%o A260885 cycbeta := PermutationGroup< 2*n |{rever}>;
%o A260885 Cbetarev := sub<G|[Cbeta,cycbeta]>;
%o A260885 psifct := function(per);
%o A260885 perinv:=per^(-1);
%o A260885 res:= [IsOdd(j) select (j+1)^per  else j-1 + 2*n : j in [1..2*n] ];
%o A260885 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 A260885 res cat:= resbis;
%o A260885 return doubleG!res;
%o A260885 end function;
%o A260885 numberofcycles := function(per);   ess :=   CycleStructure(per); return &+[ess[i,2]: i in [1..#ess]]; end function;
%o A260885 supernumberofcycles := function(per); return  numberofcycles(psifct(per)) ; end function;
%o A260885 // result given as a list genuslist (n+2-2g)^^multiplicity where g is the genus
%o A260885 // Case OU
%o A260885 dbl, dblsize := DoubleCosetRepresentatives(G,Hcycrho,Cbeta); #dblsize;
%o A260885 genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist;
%o A260885 quit;
%o A260885 // _Robert Coquereaux_, Nov 23 2015
%Y A260885 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 A260885 nonn,tabl,hard
%O A260885 1,2
%A A260885 _Robert Coquereaux_, Aug 02 2015