cp's OEIS Frontend

A057005 Number of conjugacy classes of subgroups of index n in free group of rank 2.

%I A057005 #75 Apr 04 2023 07:44:24
%S A057005 1,3,7,26,97,624,4163,34470,314493,3202839,35704007,433460014,
%T A057005 5687955737,80257406982,1211781910755,19496955286194,333041104402877,
%U A057005 6019770408287089,114794574818830735,2303332664693034476,48509766592893402121,1069983257460254131272
%N A057005 Number of conjugacy classes of subgroups of index n in free group of rank 2.
%C A057005 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
%C A057005 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
%C A057005 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.
%D A057005 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.
%H A057005 Alois P. Heinz, <a href="/A057005/b057005.txt">Table of n, a(n) for n = 1..450</a>
%H A057005 M. Deryagina, <a href="ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v446/p031.pdf">On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices</a>, Preprint 2016.
%H A057005 J. B. Geloun and S. Ramgoolam, <a href="http://arxiv.org/abs/1307.6490">Counting Tensor Model Observables and Branched Covers of the 2-Sphere</a>, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
%H A057005 Paawan Jethva, <a href="https://vrs.amsi.org.au/wp-content/uploads/sites/92/2023/03/jethva_paawan_vrs-report.pdf">Exploring the Euler Characteristics of Dessins d'Enfants</a>, Univ. Adelaide (Australia, 2023).
%H A057005 G. A. Jones and D. Singerman, <a href="http://dx.doi.org/10.1112/plms/s3-37.2.273">Theory of maps on orientable surfaces</a>, Proc. London Math. Soc. (3) 37:2 (1978), 273-307.
%H A057005 J. H. Kwak and J. Lee, <a href="https://doi.org/10.1002/(SICI)1097-0118(199610)23:2&lt;105::AID-JGT1&gt;3.0.CO;2-X">Enumeration of connected graph coverings</a>, J. Graph Th., 23 (1996), 105-109.
%H A057005 J. H. Kwak and J. Lee, <a href="http://com2mac.postech.ac.kr/resorce/Lecture_text.htm">Enumeration of graph coverings and surface branched coverings</a>, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
%H A057005 V. A. Liskovets, <a href="http://dx.doi.org/10.1023/A:1005950823566">Reductive enumeration under mutually orthogonal group actions</a>, Acta Applic. Math., 52 (1998), 91-120.
%H A057005 Carlos I. Pérez-Sánchez, <a href="https://arxiv.org/abs/1608.08134">The full Ward-Takahashi Identity for colored tensor models</a>, arXiv preprint arXiv:1608.08134 [math-ph], 2016.
%H A057005 M. Planat, A. Giorgetti, F. Holweck, and M. Saniga, <a href="https://arxiv.org/abs/1310.4267">Quantum contextual finite geometries from dessins d'efants</a>, arXiv:1310.4267 [quant-ph], 2013-2015.
%H A057005 P. Vrana, <a href="https://doi.org/10.1088/1751-8113/44/22/225304">On the algebra of local unitary invariants of pure and mixed quantum states</a>, J. Phys A: Math. Theor. 44 (2011) 225304 doi:10.1088/1751-8113/44/22/225304, Table 2.
%H A057005 Timothy R. Walsh, <a href="http://www.info2.uqam.ca/~walsh_t/papers/GENERATING NONISOMORPHIC.pdf">Space-efficient generation of nonisomorphic maps and hypermaps</a>
%H A057005 Timothy R. Walsh, <a href="http://dx.doi.org/10.1137/0604018">Generating nonisomorphic maps without storing them</a>, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
%H A057005 Timothy R. Walsh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Walsh/walsh3.html">Space-Efficient Generation of Nonisomorphic Maps and Hypermaps</a>, J. Int. Seq. 18 (2015) # 15.4.3.
%H A057005 L. Zapponi, <a href="http://www.ams.org/notices/200307/index.html">What is a dessin d'enfant?</a>, Notices AMS, 50:7, 2003, 788-789.
%H A057005 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F A057005 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
%F A057005 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
%t A057005 f[1] = {a[0] -> 0, a[1] -> 1};
%t A057005 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}];
%t A057005 s = Series[p1 - p2 /. f[max - 1], {x, 0, max}] // Normal // Expand;
%t A057005 sol = Thread[CoefficientList[s, x] == 0] // Solve // First;
%t A057005 Join[f[max - 1], sol]);
%t A057005 Array[a, 22] /. f[22] (* _Jean-François Alcover_, Mar 11 2014, updated Jan 01 2021 *)
%Y A057005 Cf. A057004-A057013. Inverse Euler transform of A110143.
%K A057005 nonn
%O A057005 1,2
%A A057005 _N. J. A. Sloane_, Sep 09 2000
%E A057005 More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001