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 A260848 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 oriented). 6
 1, 2, 1, 6, 6, 1, 21, 64, 36, 0, 99, 559, 772, 108, 0, 588, 5656, 14544, 7222, 0, 0, 3829, 56528, 246092, 277114, 34680, 0, 0, 27404, 581511, 3900698, 8180123, 3534038, 0, 0, 0, 206543, 6020787, 58838383, 203964446, 198551464, 22521600, 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 A008987 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 and J.-B. Zuber, Maps, immersions and permutations, arXiv:1507.03163 [math.CO], 2015, Table 9. EXAMPLE The transposed triangle starts: 1    2   6   21    99     588      3829      27404            206543      1   6   64   559    5656     56528     581511           6020787          1   36   772   14544    246092    3900698          58838383               0   108    7222    277114    8180123         203964446                     0      0      34680    3534038         198551464                            0          0         0           22521600                                       0         0                  0                                                 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;  // 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; 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 UO dbl, dblsize := DoubleCosetRepresentatives(G, H, 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 A260847. Cf. A008987, A260285, A260885, A260914. Sequence in context: A046521 A104684 A060538 * A110183 A110098 A244888 Adjacent sequences:  A260845 A260846 A260847 * A260849 A260850 A260851 KEYWORD nonn,tabl,hard AUTHOR Robert Coquereaux, Aug 01 2015 STATUS approved

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)