A268567
Number of immersions of oriented circle into oriented sphere with n labeled double points.
Original entry on oeis.org
1, 4, 42, 780, 21552, 803760, 38054160, 2194345440, 149604053760, 11794431720960, 1056927459571200, 106197377365094400, 11831983152533913600, 1448373107312190259200, 193295730632526147225600, 27939600631552720061952000, 4349183288219555957563392000
Offset: 1
A268563
Number of immersions of unoriented circle into oriented sphere with 2n double points such that two different choices of orientation of the circle are related by geotopy.
Original entry on oeis.org
1, 1, 5, 33, 249, 2036, 17572, 157676, 1457037
Offset: 0
A264755
Triangle T(n,g) read by rows: Partition of the set of (2n-1)! circular permutations on 2n elements according to the minimal genus g of the surface in which one can immerse the non-simple closed curves with n crossings determined by those permutations.
Original entry on oeis.org
1, 4, 2, 42, 66, 12, 780, 2652, 1608, 21552, 132240, 183168, 25920, 803760, 7984320, 20815440, 10313280
Offset: 1
Taking n = 5 crossings and genus g=0, one obtains a subset of T(5, 0) = 21552 circular permutations of Sym(10) which correspond, in the OO case (the circle is oriented, the sphere is oriented), to the union 179 orbits of length 120=5!/1 and 3 orbits of length 24=5!/5 with respective centralizers of order 1 and 5 under the action of the symmetric group Sym(5) acting on this subset: 179*120 + 3*24 = 21552. The total number of orbits 179 + 3 = 182 = A008986(5) = A260285(5, 0) is the number of immersed spherical curves (g=0) with 5 crossings, in the OO case. The next entry, T(5, 1) = 132240, gives the number of circular permutations that describe immersed closed curves in a torus (g=1), with n=5 crossings, up to stable geotopy; the number of such closed curves in the OO case is 1102 = A260285(5, 1).
Triangle begins:
1
4 2
42 66 12
780 2652 1608
21552 132240 183168 25920
803760 7984320 20815440 10313280
...
-
/* Example: line n=5 of the triangle */
n:=5;
G:=Sym(2*n);
CG := Classes(G);
pos:= [j: j in [1..#CG] | CycleStructure(CG[j][3]) eq [<2*n,1>]][1];
circularpermutations:=Class(G,CG[pos][3]); //circularpermutations
doubleG := Sym(4*n);
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;
{* supernumberofcycles(x) : x in circularpermutations *};
quit;
A268564
Number of immersions of unoriented circle into oriented sphere with 2n double points such that two different choices of orientation of the circle are not related by geotopy.
Original entry on oeis.org
0, 1, 16, 555, 27155, 1624602, 110848296, 8306916404, 667683029392
Offset: 0
A268565
Number of achiral immersions of oriented circle into oriented sphere with 2n double points.
Original entry on oeis.org
1, 1, 5, 15, 97, 592, 4488, 35314, 295379
Offset: 0
A268566
Number of chiral immersions of oriented circle into oriented sphere with 2n double points.
Original entry on oeis.org
0, 1, 16, 564, 27231, 1625324, 110854838, 8306977585, 667683610221
Offset: 0
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