The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A260885 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 oriented, the surface is unoriented). 6
 1, 2, 1, 6, 6, 2, 21, 62, 37, 0, 97, 559, 788, 112, 0, 579, 5614, 14558, 7223, 0, 0, 3812, 56526, 246331, 277407, 34748, 0, 0, 27328, 580860, 3900740, 8179658, 3534594, 0, 0, 0, 206410, 6020736, 58842028, 203974134, 198559566, 22524176, 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 A008988 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, J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474 EXAMPLE The transposed triangle starts: 1 2 6 21 97 579 3812 27328 206410 1 6 62 559 5614 56526 580860 6020736 2 37 788 14558 246331 3900740 58842028 0 112 7223 277407 8179658 203974134 0 0 34748 3534594 198559566 0 0 0 22524176 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 OU dbl, dblsize := DoubleCosetRepresentatives(G, Hcycrho, Cbeta); #dblsize; genuslist := {* supernumberofcycles(beta^(dbl[j]^(-1))) : j in [1..#dblsize] *}; genuslist; quit; # Robert Coquereaux, Nov 23 2015 CROSSREFS Cf. A008988. The sum over all genera g for a fixed number n of crossings is given by sequence A260887. Cf. A260885, A260848, A260914. Sequence in context: A232433 A271881 A182729 * A075181 A052121 A193895 Adjacent sequences: A260882 A260883 A260884 * A260886 A260887 A260888 KEYWORD nonn,tabl,hard AUTHOR Robert Coquereaux, Aug 02 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 23 20:39 EDT 2023. Contains 361452 sequences. (Running on oeis4.)