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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. 5
1, 4, 22, 218, 3028, 55540, 1235526, 32434108, 980179566, 33522177088, 1279935820810, 53970628896500, 2490952020480012, 124903451391713412, 6761440164391403896, 393008709559373134184, 24412776311194951680016, 1613955767240361647220648, 113146793787569865523200018, 8384177419658944198600637096 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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 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

CROSSREFS

Cf. A260285, A260912, A260847, A260887.

Sequence in context: A207156 A197999 A197963 * A302769 A137158 A025135

Adjacent sequences:  A260293 A260294 A260295 * A260297 A260298 A260299

KEYWORD

nonn

AUTHOR

Robert Coquereaux, Jul 22 2015

STATUS

approved

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Last modified March 4 03:53 EST 2021. Contains 341779 sequences. (Running on oeis4.)