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A260887 Sum over the genera g of the number of immersions of an oriented circle with n crossings in an unoriented surface of genus g. 4
1, 3, 14, 120, 1556, 27974, 618824, 16223180, 490127050, 16761331644, 639969571892, 26985326408240, 1245476099801252, 62451726395242858, 3380720087847928728, 196504354827002278248, 12206388156005725243280, 806977883623811932432386, 56573396893818112613554940, 4192088709829783508863131872 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is the sum over the n-th row of the triangle A260885.

a(n) is also the number of double cosets of H\G/K where G is the symmetric group S(2n), H is the subgroup generated by the centralizer of the circular permutation β = (1,2,3,...,2n) of G, K is a subgroup of G generated by the permutation ρ = (1,2)(3,4)...(2n-3,2n-2)(2n-1,2n), using cycle notation, and the subgroup (isomorphic with S(n)) that commutes with ρ and permutes odd resp. even integers among themselves.

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

Table of n, a(n) for n=1..20.

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

PROG

(MAGMA) /* For all n */

nbofdblecos := function(G, H, K);

CG := Classes(G); nCG := #CG; oG := #G; CH := Classes(H); nCH := #CH; oH := #H; CK := Classes(K); nCK := #CK; oK := #K;

resH := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Hmupositions := {j: j in [1..nCH]  | CycleStructure(CH[j][3]) eq CycleStructure(Gmurep)};

Hmugoodpositions := {j : j in Hmupositions | IsConjugate(G, CH[j][3], Gmurep) eq true}; bide := 0; for j in Hmugoodpositions do bide := bide + CH[j][2]; end for;

Append(~resH, bide); end for;

resK := []; for mu in [1..nCG] do  Gmurep := CG[mu][3]; Kmupositions := {j: j in [1..nCK]  | CycleStructure(CK[j][3]) eq CycleStructure(Gmurep)};

Kmugoodpositions := {j : j in Kmupositions | IsConjugate(G, CK[j][3], Gmurep) eq true}; bide := 0; for j in Kmugoodpositions do bide := bide + CK[j][2]; end for;

Append(~resK, bide); end for;

ndcl := 0; tot := 0; for mu in [1..nCG] do tot := tot + resH[mu]* resK[mu]/CG[mu][2]; end for;  ndcl:= tot *  oG/(oH * oK); return ndcl;

end function;

OUfull := function(n); G:=Sym(2*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>;

beta:=G!Append([2..2*n], 1); Cbeta:=Centralizer(G, beta);

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}>;  Hcycrho:=sub<G|[H, cycrho]>;

return nbofdblecos(G, Hcycrho, Cbeta); end function;

[OUfull(n) : n in [1..10]]; //

CROSSREFS

Cf. A260296, A260847, A260885, A260912.

Sequence in context: A007140 A161936 A304983 * A127850 A324147 A186772

Adjacent sequences:  A260884 A260885 A260886 * A260888 A260889 A260890

KEYWORD

nonn

AUTHOR

Robert Coquereaux, Aug 02 2015

STATUS

approved

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Last modified September 15 08:52 EDT 2019. Contains 327062 sequences. (Running on oeis4.)