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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.

%I A264755 #37 Mar 05 2025 05:25:50
%S A264755 1,4,2,42,66,12,780,2652,1608,21552,132240,183168,25920,803760,
%T A264755 7984320,20815440,10313280
%N 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.
%C A264755 Each line of the triangle adds up to an odd factorial (2n-1)!. Example (line n=5): 21552 + 132240 + 183168 + 25920 = 362880 = 9!.
%C A264755 The lengths of the rows of the triangle do not strictly increase with n, the first lengths are (1,2,3,3,4,4,...).
%H A264755 R. Coquereaux and J.-B. Zuber, <a href="http://arxiv.org/abs/1507.03163">Maps, immersions and permutations</a>, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: <a href="https://doi.org/10.1142/S0218216516500474">10.1142/S0218216516500474</a>
%e A264755 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).
%e A264755 Triangle begins:
%e A264755   1
%e A264755   4 2
%e A264755   42 66 12
%e A264755   780 2652 1608
%e A264755   21552 132240 183168 25920
%e A264755   803760 7984320 20815440 10313280
%e A264755   ...
%o A264755 (Magma) /* Example: line n=5 of the triangle */
%o A264755 n:=5;
%o A264755 G:=Sym(2*n);
%o A264755 CG := Classes(G);
%o A264755 pos:= [j: j in [1..#CG]  | CycleStructure(CG[j][3]) eq [<2*n,1>]][1];
%o A264755 circularpermutations:=Class(G,CG[pos][3]); //circularpermutations
%o A264755 doubleG := Sym(4*n);
%o A264755 psifct := function(per);
%o A264755 perinv:=per^(-1);
%o A264755 res:= [IsOdd(j) select (j+1)^per  else j-1 + 2*n : j in [1..2*n] ];
%o A264755 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] ];
%o A264755 res cat:= resbis;
%o A264755 return doubleG!res;
%o A264755 end function;
%o A264755 numberofcycles := function(per);   ess :=   CycleStructure(per); return &+[ess[i,2]: i in [1..#ess]]; end function;
%o A264755 supernumberofcycles := function(per); return  numberofcycles(psifct(per)) ; end function;
%o A264755 {* supernumberofcycles(x) : x in circularpermutations  *};
%o A264755 quit;
%Y A264755 Cf. A008986, A260296.
%Y A264755 Cf. A260285, A260848, A260885, A260914, A268567 (g=0).
%K A264755 nonn,tabf,more
%O A264755 1,2
%A A264755 _Robert Coquereaux_, Nov 23 2015