A057013 -id:A057013 - cp's OEIS frontend

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

1, 3, 7, 26, 97, 624, 4163, 34470, 314493, 3202839, 35704007, 433460014, 5687955737, 80257406982, 1211781910755, 19496955286194, 333041104402877, 6019770408287089, 114794574818830735, 2303332664693034476, 48509766592893402121, 1069983257460254131272

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

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.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

Alois P. Heinz, Table of n, a(n) for n = 1..450
M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.
J. B. Geloun and S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
Paawan Jethva, Exploring the Euler Characteristics of Dessins d'Enfants, Univ. Adelaide (Australia, 2023).
G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37:2 (1978), 273-307.
J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.
Carlos I. Pérez-Sánchez, The full Ward-Takahashi Identity for colored tensor models, arXiv preprint arXiv:1608.08134 [math-ph], 2016.
M. Planat, A. Giorgetti, F. Holweck, and M. Saniga, Quantum contextual finite geometries from dessins d'efants, arXiv:1310.4267 [quant-ph], 2013-2015.
P. Vrana, On the algebra of local unitary invariants of pure and mixed quantum states, J. Phys A: Math. Theor. 44 (2011) 225304 doi:10.1088/1751-8113/44/22/225304, Table 2.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Generating nonisomorphic maps without storing them, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 161-178.
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
L. Zapponi, What is a dessin d'enfant?, Notices AMS, 50:7, 2003, 788-789.
Index entries for sequences related to groups

Cf. A057004-A057013. Inverse Euler transform of A110143.

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]);
Array[a, 22] /. f[22] (* Jean-François Alcover, Mar 11 2014, updated Jan 01 2021 *)

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

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A057004 Array T(n,k) = number of conjugacy classes of subgroups of index k in free group of rank n, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 41, 26, 1, 1, 31, 235, 604, 97, 1, 1, 63, 1361, 14120, 13753, 624, 1, 1, 127, 7987, 334576, 1712845, 504243, 4163, 1, 1, 255, 47321, 7987616, 207009649, 371515454, 24824785, 34470, 1, 1, 511, 281995, 191318464

Offset: 1

N. J. A. Sloane, Sep 09 2000

			Array T(n,k) begins:
1 1 1 1 1 1 1 ...
1 3 7 26 97 624 4163 ...
1 7 41 604 13753 504243 ...
1 15 235 14120 1712845 ...

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

M. Hofmeister, A Note on Counting Connected Graph Covering Projections, SIAM J. Discrete Math., 11 (1998), 286-292. See page 291 Table 4.3.
J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.

Rows, columns, main diagonal give A057005-A057013, A160871.

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A057006 Number of conjugacy classes of subgroups of index n in free group of rank 3.

Original entry on oeis.org

1, 7, 41, 604, 13753, 504243, 24824785, 1598346352, 129958211233, 13030565312011, 1579721338432537, 227804599861102676, 38541084552054952009, 7560534755192908672087, 1702288146755359962223409

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

Vaclav Kotesovec, Table of n, a(n) for n = 1..60
J. B. Geloun and S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
Joseph Ben Geloun and Sanjaye Ramgoolam, All-orders asymptotics of tensor model observables from symmetries of restricted partitions, arXiv:2106.01470 [hep-th], 2021.
J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.
P. Vrana, On the algebra of local unitary invariants of pure and mixed quantum states, J. Phys A: Math. Theor. 44 (2011) 225304 doi:10.1088/1751-8113/44/22/225304, Table 2.

Cf. A057004-A057013. Inverse Euler transform of A152612.

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A057007 Number of conjugacy classes of subgroups of index n in free group of rank 4.

Original entry on oeis.org

1, 15, 235, 14120, 1712845, 371515454, 127635996839, 65417303558808, 47718183491375681, 47736450176936606373, 63553335359217380046467, 109839393591932655104274298, 241347279732331127743077516005

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

Vaclav Kotesovec, Table of n, a(n) for n = 1..60
J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.
P. Vrana, On the algebra of local unitary invariants of pure and mixed quantum states, J. Phys A: Math. Theor. 44 (2011) 225304 doi:10.1088/1751-8113/44/22/225304, Table 2.

Cf. A057004-A057013. Inverse Euler transform of A160446.

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A057012 Number of conjugacy classes of subgroups of index 6 in free group of rank n.

Original entry on oeis.org

1, 624, 504243, 371515454, 268530771271, 193466859054994, 139311082645798043, 100305771690618678654, 72220370631411094037391, 51998692654400641678907114, 37439061807069469917891862243

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.

Cf. A057004-A057013.

a(n)=if(n<0,0,n--;720^n-120^n+24^n+18^n-16^n+12^n*2-36^n/2-9^n/2+8^n*2/3-6^n/2-4^n*3/2-3^n/2+2^n*5/6)

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A057008 Number of conjugacy classes of subgroups of index n in free group of rank 5.

Original entry on oeis.org

1, 31, 1361, 334576, 207009649, 268530771271, 644969015852641, 2642258726261312896, 17337468317813249073281, 173383832437017181584973783, 2538593496477071922439308897841, 52641334467700181691498204296713016, 1503509056221581488501100896035227575441

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.

Cf. A057004-A057013.

Inverse Euler transform of A160447.

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A057010 Number of conjugacy classes of subgroups of index 4 in free group of rank n.

Original entry on oeis.org

1, 26, 604, 14120, 334576, 7987616, 191318464, 4588288640, 110090411776, 2641931680256, 63404394241024, 1521689370306560, 36520413978750976, 876488875160477696, 21035724442850934784, 504857317670233210880

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.

Cf. A057004-A057013.

PARI
```
a(n)=if(n<0,0,24^(n-1)+8^(n-1)-6^(n-1))
```

G.f.: x(1-12x)/((1-6x)(1-8x)(1-24x)).

a(n) = 24^(n-1)+8^(n-1)-6^(n-1).

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A057011 Number of conjugacy classes of subgroups of index 5 in free group of rank n.

Original entry on oeis.org

1, 97, 13753, 1712845, 207009649, 24875000437, 2985789977353, 358313458071085, 42998059096839649, 5159777705044971877, 619173578774772949753, 74300835546376264277725

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.

Cf. A057004-A057013.

a(n)=if(n<0,0,n--;120^n-24^n-12^n+6^n+5^n+4^n-2^n)

G.f.: x(1-76x+4336x^2-81504x^3+522720x^4-1064448x^5)/((1-2x)(1-4x)(1-5x)(1-6x)(1-12x)(1-24x)(1-120x)).

a(n) = 120^(n-1)-24^(n-1)-12^(n-1)+6^(n-1)+5^(n-1)+4^(n-1)-2^(n-1).

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A057009 Number of conjugacy classes of subgroups of index 3 in free group of rank n.

Original entry on oeis.org

1, 7, 41, 235, 1361, 7987, 47321, 281995, 1685921, 10096867, 60524201, 362972155, 2177309681, 13062280147, 78368930681, 470199300715, 2821152888641, 16926788453827, 101560343826761, 609360901747675, 3656161925798801, 21936961098633907, 131621735219132441

Offset: 1

N. J. A. Sloane, Sep 09 2000

nonn

Starting at a(2), consider that 2/3 - 1/2 = 1/6 with 1+6=7=a(2); 8/9 - 3/4 = 5/36 with 5+36=41=a(3); 26/27 - 7/8=19/216 with 19+216=235=a(4); 80/81 - 15/16=65/1296 with 65+1296=1361=a(5) and so forth. The numerators starting at a(3) are 5,19,65,211,665,2059,6305,... (see A001047) with 19 mod 5=4, 65 mod 19=8, 211 mod 65=16, 665 mod 211=32, 2059 mod 665=64, 6305 mod 2059=128, and so forth for higher powers of 2. - J. M. Bergot, May 09 2015

In other words, let f(n) = (3^(n-1)-1)/3^(n-1) - (2^(n-1)-1)/2^(n-1), then for n>=1 a(n) = numerator(f(n)) + denominator(f(n)). - Michel Marcus, May 29 2015

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(c), pp. 76, 112.

J. H. Kwak and J. Lee, Enumeration of connected graph coverings, J. Graph Th., 23 (1996), 105-109.
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3.
V. A. Liskovets, Reductive enumeration under mutually orthogonal group actions, Acta Applic. Math., 52 (1998), 91-120.
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Cf. A057004, A057005, A057006, A057007, A057008, A057010, A057011, A057012, A057013.

[6^(n-1)+3^(n-1)-2^(n-1): n in [1..30]] /* or */ I:=[1,7,41]; [n le 3 select I[n] else 11*Self(n-1)-36*Self(n-2)+36*Self(n-3): n in [1..30]]; // Vincenzo Librandi, May 12 2015

Table[6^(n-1)+3^(n-1)-2^(n-1),{n,25}] (* or *) LinearRecurrence[ {11,-36,36},{1,7,41},25] (* Harvey P. Dale, Nov 24 2011 *)
CoefficientList[Series[(1 - 4 x)/((1 - 2 x) (1 - 3 x) (1 - 6 x)), {x, 0, 33}], x] (* Vincenzo Librandi, May 12 2015 *)

PARI
```
a(n)=if(n<0,0,6^(n-1)+3^(n-1)-2^(n-1))
```

G.f.: x(1-4x)/((1-2x)(1-3x)(1-6x)). a(n) = 6^(n-1)+3^(n-1)-2^(n-1).
E.g.f.: e^(6*x)+e^(3*x)-e^(2*x). [Mohammad K. Azarian, Jan 16 2009]
a(1)=1, a(2)=7, a(3)=41, a(n) = 11*a(n-1)-36*a(n-2)+36*a(n-3). [Harvey P. Dale, Nov 24 2011]

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

cp's OEIS Frontend

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

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A057004 Array T(n,k) = number of conjugacy classes of subgroups of index k in free group of rank n, read by antidiagonals.

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A057006 Number of conjugacy classes of subgroups of index n in free group of rank 3.

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A057007 Number of conjugacy classes of subgroups of index n in free group of rank 4.

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A057012 Number of conjugacy classes of subgroups of index 6 in free group of rank n.

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A057008 Number of conjugacy classes of subgroups of index n in free group of rank 5.

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A057010 Number of conjugacy classes of subgroups of index 4 in free group of rank n.

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A057011 Number of conjugacy classes of subgroups of index 5 in free group of rank n.

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A057009 Number of conjugacy classes of subgroups of index 3 in free group of rank n.

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