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%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
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