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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). 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 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

LINKS

Table of n, a(n) for n=1..45.

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

EXAMPLE

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

PROG

(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<G|[H, cycrho]>;  // 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<G|[Cbeta, cycbeta]>;

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

CROSSREFS

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

Cf. A008987, A260285, A260885, A260914.

Sequence in context: A046521 A104684 A060538 * A110183 A110098 A244888

Adjacent sequences:  A260845 A260846 A260847 * A260849 A260850 A260851

KEYWORD

nonn,tabl,hard

AUTHOR

Robert Coquereaux, Aug 01 2015

STATUS

approved

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)