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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
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
KEYWORD
nonn,hard,tabl
AUTHOR
Robert Coquereaux, Aug 04 2015
STATUS
approved

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Last modified March 28 13:19 EDT 2024. Contains 371254 sequences. (Running on oeis4.)