|
|
A057005
|
|
Number of conjugacy classes of subgroups of index n in free group of rank 2.
|
|
13
|
|
|
1, 3, 7, 26, 97, 624, 4163, 34470, 314493, 3202839, 35704007, 433460014, 5687955737, 80257406982, 1211781910755, 19496955286194, 333041104402877, 6019770408287089, 114794574818830735, 2303332664693034476, 48509766592893402121, 1069983257460254131272
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Number of (unlabeled) dessins d'enfants with n edges. A dessin d'enfant ("child's drawing") by A. Grothendieck, 1984, is a connected bipartite multigraph with properly bicolored nodes (w and b) in which a cyclic order of the incident edges is assigned to every node. For n=2 these are w--b--w, b--w--b and w==b. - Valery A. Liskovets, Mar 17 2005
Also (apparently), a(n+1) = number of sensed hypermaps with n darts on a surface of any genus (see Walsh 2012). - N. J. A. Sloane, Aug 01 2012
Response from Timothy R. Walsh, Aug 01 2012: The conjecture in the previous comment is true. A combinatorial map is a connected graph, loops and multiple edges allowed, in which a cyclic order of the incident edge-ends is assigned to every node. The equivalence between combinatorial maps and topological maps was conjectured by several researchers and finally proved by Jones and Singerman. In my 1975 paper "Generating nonisomorphic maps without storing them", I established a genus-preserving bijection between hypermaps with n darts, w vertices and b edges and properly bicolored bipartite maps with n edges, w white vertices and b black vertices. A bipartite map can't have any loops; so a combinatorial bipartite map is a multigraph and it suffices to impose a cyclic order of the edges, rather than the edge-ends, incident to each node. Thus it is just the child's drawing described above by Liskovets.
|
|
REFERENCES
|
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.
|
|
LINKS
|
|
|
FORMULA
|
prod_{n>0} (1-x^n)^{-a(n)} = prod_{i>0} sum_{j>=0} j!*i^j*x^{i*j}. (Liskovets) - Valery A. Liskovets, Mar 17 2005 ... and both sides = sum_{j>=0} A110143(j)*x^j . - R. J. Mathar, Oct 18 2012
a(n) ~ n! * (1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - 171/n^5 - 1542/n^6 - 16241/n^7 - 194973/n^8 - 2622610/n^9 - 39027573/n^10 - ...), for the coefficients see A113869. - Vaclav Kotesovec, Aug 09 2019
|
|
MATHEMATICA
|
f[1] = {a[0] -> 0, a[1] -> 1};
f[max_] := f[max] = (p1 = Product[(1 - x^n)^(-a[n]), {n, 0, max}]; p2 = Product[Sum[j!*If[j == 0, 1, i^j]*x^(i*j), {j, 0, max}], {i, 0, max}];
s = Series[p1 - p2 /. f[max - 1], {x, 0, max}] // Normal // Expand;
sol = Thread[CoefficientList[s, x] == 0] // Solve // First;
Join[f[max - 1], sol]);
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001
|
|
STATUS
|
approved
|
|
|
|