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.

1, 4, 2, 42, 66, 12, 780, 2652, 1608, 21552, 132240, 183168, 25920, 803760, 7984320, 20815440, 10313280

Offset

1,2

Comments

Each line of the triangle adds up to an odd factorial (2n-1)!. Example (line n=5): 21552 + 132240 + 183168 + 25920 = 362880 = 9!.

The lengths of the rows of the triangle do not strictly increase with n, the first lengths are (1,2,3,3,4,4,...).

Links

Table of n, a(n) for n=1..17.

R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: 10.1142/S0218216516500474

Example

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
  ...

Prog

(Magma) /* 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;

Crossrefs

Cf. A008986, A260296.

Cf. A260285, A260848, A260885, A260914, A268567 (g=0).

Sequence in context: A303625 A123850 A280780 * A120968 A193894 A107667

Adjacent sequences: A264752 A264753 A264754 * A264756 A264757 A264758

Keyword

nonn,tabf,more

Author

Robert Coquereaux, Nov 23 2015

Status

approved