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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. 0
1, 4, 2, 42, 66, 12, 780, 2652, 1608, 21552, 132240, 183168, 25920, 803760, 7984320, 20815440, 10313280 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A303625 A123850 A280780 * A120968 A193894 A107667
KEYWORD
nonn,tabf,more
AUTHOR
Robert Coquereaux, Nov 23 2015
STATUS
approved

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Last modified July 12 08:07 EDT 2024. Contains 374239 sequences. (Running on oeis4.)