A260847 Sum over the genera g of the number of immersions of an unoriented circle with n crossing in an oriented surface of genus g.

1, 3, 13, 121, 1538, 28010, 618243, 16223774, 490103223, 16761330464, 639968394245, 26985325092730, 1245476031528966, 62451726249369666, 3380720083302727868, 196504354812897344692, 12206388155663897395208, 806977883622439156487124, 56573396893789449427353609, 4192088709829643732598955348

Offset: 1

Author

Robert Coquereaux, Aug 01 2015

Keywords

• nonn

Comments

a(n) is the sum over the n-th row of triangle A260848.

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, and K is a subgroup of G isomorphic with S(n) that commutes with

(1,2)(3,4)...(2n-3,2n-2)(2n-1,2n), using cycle notation, 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

• 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
• R. J. Mathar, Chord Diagrams with Directed Chords, viXra:2410.0145 (2024)

Crossrefs

Cf. A260848, A260296, A260912, A260887.

Programs

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;
UOfull := 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; return nbofdblecos(G,H,Cbetarev); end function;
[UOfull(n) : n in [1..10]]; //