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).

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

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