A260912 Sum over the genera g of the number of immersions of an unoriented circle with n crossings in an unoriented surface of genus g.

1, 3, 12, 86, 894, 14715, 313364, 8139398, 245237925, 8382002270, 319994166042, 13492740284184, 622738642693202, 31225868370080949, 1690360086869176780, 98252177808632109236, 6103194081506193327048, 403488941845715112039425, 28286698447226523233226110, 2096044354918091666701275248

Offset

1,2

Comments

a(n) is the sum over the n-th row of triangle A260914.

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 together with the permutation (2, 2n)(3, 2n-1)(4, 2n-2)...(n, n+2) that conjugates β and β-1, 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;
UUfull := 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); bool, rever := IsConjugate(G, beta, beta^(-1)); cycbeta := PermutationGroup< 2*n |{rever}>;
Cbetarev := sub<G|[Cbeta, cycbeta]>;
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, Cbetarev); end function;
[UUfull(n) : n in [1..10]]; //

Crossrefs

Cf. A260847, A260887, A260296, A260914.

Sequence in context: A232934 A077047 A074505 * A266788 A039305 A174463

Adjacent sequences: A260909 A260910 A260911 * A260913 A260914 A260915

Keyword

nonn

Author

Robert Coquereaux, Aug 04 2015

Status

approved