A260296 Sum over the genera g of the number of immersions of an oriented circle with n crossing in an oriented surface of genus g.

1, 4, 22, 218, 3028, 55540, 1235526, 32434108, 980179566, 33522177088, 1279935820810, 53970628896500, 2490952020480012, 124903451391713412, 6761440164391403896, 393008709559373134184, 24412776311194951680016, 1613955767240361647220648, 113146793787569865523200018, 8384177419658944198600637096

Offset: 1

Author

Robert Coquereaux, Jul 22 2015

Keywords

• nonn

Comments

a(n) is the sum over the n-th row of triangle A260285.

a(n) is also the number of double cosets of H\G/K where G is the symmetric group S(2n), H is the centralizer of a circular permutation of G, 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 n a prime integer, there is an explicit formula: a(n) = n-1 +(2n-1)!/n!.

For given g > 0 the immersions are understood up to stable geotopy equivalence (listed 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
• R. J. Mathar, Chord Diagrams with Directed Chords, viXra:2410.0145 (2024)

Crossrefs

Cf. A260285, A260912, A260847, A260887.

Programs

Magma

/* For n a prime integer */ [NthPrime(n)-1 +Factorial(2*NthPrime(n)-1) div Factorial(NthPrime(n)): n in [0..10]]; // Vincenzo Librandi, Aug 01 2015

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;
OOfull := 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);
return nbofdblecos(G, H, Cbeta); end function;
[OOfull(n) : n in [1..10]];
// Robert Coquereaux, Aug 01 2015