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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). 6
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 (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 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

LINKS

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

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

EXAMPLE

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

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 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

CROSSREFS

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

Sequence in context: A008970 A055896 A193723 * A159965 A116395 A159924

Adjacent sequences:  A260911 A260912 A260913 * A260915 A260916 A260917

KEYWORD

nonn,hard,tabl

AUTHOR

Robert Coquereaux, Aug 04 2015

STATUS

approved

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Last modified February 21 09:51 EST 2018. Contains 299390 sequences. (Running on oeis4.)