%I #31 Oct 24 2024 04:36:24
%S 1,4,22,218,3028,55540,1235526,32434108,980179566,33522177088,
%T 1279935820810,53970628896500,2490952020480012,124903451391713412,
%U 6761440164391403896,393008709559373134184,24412776311194951680016,1613955767240361647220648,113146793787569865523200018,8384177419658944198600637096
%N Sum over the genera g of the number of immersions of an oriented circle with n crossing in an oriented surface of genus g.
%C a(n) is the sum over the n-th row of triangle A260285.
%C 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
%C (1,2)(3,4)...(2n-3,2n-2)(2n-1,2n), using cycle notation, and permutes odd resp. even integers among themselves.
%C For n a prime integer, there is an explicit formula: a(n) = n-1 +(2n-1)!/n!.
%C 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
%H R. Coquereaux, J.-B. Zuber, <a href="http://arxiv.org/abs/1507.03163">Maps, immersions and permutations</a>, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), <a href="http://dx.doi.org/10.1142/S0218216516500474">DOI</a>
%H R. J. Mathar, <a href="https://vixra.org/abs/2410.0145">Chord Diagrams with Directed Chords</a>, viXra:2410.0145 (2024)
%o (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
%o (Magma) /* For all n */
%o nbofdblecos := function(G, H, K);
%o CG := Classes(G); nCG := #CG; oG := #G; CH := Classes(H); nCH := #CH; oH := #H; CK := Classes(K); nCK := #CK; oK := #K;
%o resH := []; for mu in [1..nCG] do Gmurep := CG[mu][3]; Hmupositions := {j: j in [1..nCH] | CycleStructure(CH[j][3]) eq CycleStructure(Gmurep)};
%o 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;
%o resK := []; for mu in [1..nCG] do Gmurep := CG[mu][3]; Kmupositions := {j: j in [1..nCK] | CycleStructure(CK[j][3]) eq CycleStructure(Gmurep)};
%o 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;
%o 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;
%o end function;
%o 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;
%o H := PermutationGroup< 2*n |genH>;
%o beta:=G!Append([2..2*n],1); Cbeta:=Centralizer(G,beta);
%o return nbofdblecos(G, H, Cbeta); end function;
%o [OOfull(n) : n in [1..10]];
%o // _Robert Coquereaux_, Aug 01 2015
%Y Cf. A260285, A260912, A260847, A260887.
%K nonn
%O 1,2
%A _Robert Coquereaux_, Jul 22 2015