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Sum over the genera g of the number of immersions of an unoriented circle with n crossing in an oriented surface of genus g.
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%I #22 Oct 24 2024 04:34:23

%S 1,3,13,121,1538,28010,618243,16223774,490103223,16761330464,

%T 639968394245,26985325092730,1245476031528966,62451726249369666,

%U 3380720083302727868,196504354812897344692,12206388155663897395208,806977883622439156487124,56573396893789449427353609,4192088709829643732598955348

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

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

%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 subgroup generated by the centralizer of the circular permutation β = (1,2,3,...,2n)

%C 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

%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 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

%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), DOI: http://dx.doi.org/10.1142/S0218216516500474

%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 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;

%o 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;

%o 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 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;

%o H := PermutationGroup< 2*n |genH>;

%o beta:=G!Append([2..2*n],1); Cbeta:=Centralizer(G,beta); bool, rever := IsConjugate(G,beta,beta^(-1));

%o cycbeta := PermutationGroup< 2*n |{rever}>; Cbetarev := sub<G|[Cbeta,cycbeta]>; return nbofdblecos(G,H,Cbetarev); end function;

%o [UOfull(n) : n in [1..10]]; //

%Y Cf. A260848, A260296, A260912, A260887.

%K nonn

%O 1,2

%A _Robert Coquereaux_, Aug 01 2015