A118094 -id:A118094 - cp's OEIS frontend

A090371 Number of unrooted planar 2-constellations with n digons. Also number of n-edge unrooted planar Eulerian maps with bicolored faces.

1, 3, 6, 20, 60, 291, 1310, 6975, 37746, 215602, 1262874, 7611156, 46814132, 293447817, 1868710728, 12068905911, 78913940784, 521709872895, 3483289035186, 23464708686960, 159346213738020, 1090073011199451, 7507285094455566, 52021636161126702

Offset: 1

Valery A. Liskovets, Dec 01 2003

a(n) is also the number of unrooted planar hypermaps with n darts up to orientation-preserving homeomorphism (darts are semi-edges in the particular case of ordinary maps). - Valery A. Liskovets, Apr 13 2006

			The 3 Eulerian maps with 2 edges are the digon and two figure eight graphs ("8") in which both loops are colored, resp., black or white.

R. J. Mathar, Table of n, a(n) for n = 1..100
M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. v.24 (2000), 337-368.
A. Mednykh and R. Nedela, Counting unrooted hypermaps on closed orientable surface, 18th Intern. Conf. Formal Power Series & Algebr. Comb., Jun 19, 2006, San Diego, California (USA).
A. Mednykh and R. Nedela, Enumeration of unrooted hypermaps of a given genus, Discr. Math., 310 (2010), 518-526. [From _N. J. A. Sloane_, Dec 19 2009]
A. Mednykh and R. Nedela, Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 2.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.

Cf. A000257, A069727, A090372, A118094.

A090371 := proc(n)
    local s, d;
    if n=0 then
        1 ;
    else
        s := -2^n*binomial(2*n, n);
        for d in numtheory[divisors](n) do
            s := s+ numtheory[phi](n/d)*2^d*binomial(2*d, d)
        od;
        3/(2*n)*(2^n*binomial(2*n, n)/((n+1)*(n+2))+s/2);
    fi;
end proc:

h0[n_] := 3*2^(n-1)*Binomial[2*n, n]/((n+1)*(n+2)); a[n_] := (h0[n] + DivisorSum[n, If[#>1, EulerPhi[#]*Binomial[n/#+2, 2]*h0[n/#], 0]&])/n; Array[a, 30] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)

h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
a(n) = (h0(n) + sumdiv(n, d, (d>1)*eulerphi(d)*binomial(n/d+2,2)*h0(n/d)))/n; \\ Michel Marcus, Dec 11 2014

More terms from Michel Marcus, Dec 11 2014

A214819 Number of genus 2 sensed hypermaps with n darts.

Original entry on oeis.org

0, 0, 0, 0, 4, 48, 708, 9807, 119436, 1355400, 14561360, 150429819, 1506841872, 14732613116, 141226638540, 1331912032173, 12390368538412, 113927616087252, 1037080582036632, 9358430685657218, 83804192879934456, 745394788170961932, 6590038606472968276

Offset: 1

N. J. A. Sloane, Aug 01 2012

nonn

A. Mednykh and R. Nedela, Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 4.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.

Cf. A090371, A118094, A214820, A057005, A380453.

hO[d_, , ] := 0 /; !IntegerQ@d;
hO[d_, g_, q_] := Multinomial[d+2-2g-Total@q, Sequence@@q] h[g][d];
h[0][m_] := 3 2^(m-1) Binomial[2m,m] / ((m+1)(m+2));
h[1][d_] := Sum[2^k (4^(d-2-k)-1) Binomial[d+k,k], {k,0,d-3}] / 3;
h[2][d_] := Coefficient[-# (# - 1)^5 (#^4 - 6 #^3 + 36 #^2 - 50 # + 51) / (4 (# - 2)^7 (# + 1)^5) &[(1-Sqrt[1-8x])/(4x)+O[x]^(d+1)], x, d];
a2[d_] := (h[2][d] + 4hO[d/2,1,{2}] + hO[d/2,0,{6}] + 6hO[d/3,0,{0,4}] + 2hO[d/4,0,{2,0,2}] + 12hO[d/5,0,{0,0,0,3}] + 2hO[d/6,0,{2,2}] + 2hO[d/6,0,{0,1,0,0,2}] + 4hO[d/8,0,{1,0,0,0,0,0,2}] + 4hO[d/10,0,{1,0,0,1,0,0,0,0,1}]) / d;
Table[a2[n], {n, 23}] (* Andrei Zabolotskii, Jun 24 2025, using Mednykh & Nedela's Theorem 8 *)

Terms a(13) onwards from Andrei Zabolotskii, Jun 24 2025

A214820 Number of genus 3 sensed hypermaps with n darts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 30, 1155, 29910, 601364, 10260804, 156469887, 2195431068, 28897471080, 361514582340, 4339280187364, 50323775391144, 566914469842923, 6229721664499224, 67000302262906866, 707159710965012834, 7341038807584085816, 75093327553430134548

Offset: 1

N. J. A. Sloane, Aug 01 2012

nonn

A. Mednykh and R. Nedela, Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 5.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
Timothy R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.

Cf. A090371, A118094, A214819, A057005, A380453.

hO[d_, , ] := 0 /; !IntegerQ@d;
hO[d_, g_, q_] := Multinomial[d+2-2g-Total@q, Sequence@@q] h[g][d];
h[0][m_] := 3 2^(m-1) Binomial[2m, m] / ((m+1)(m+2));
h[1][d_] := Sum[2^k (4^(d-2-k)-1) Binomial[d+k, k], {k, 0, d-3}] / 3;
h[2][d_] := Coefficient[-# (# - 1)^5 (#^4 - 6 #^3 + 36 #^2 - 50 # + 51) / (4 (# - 2)^7 (# + 1)^5) &[(1-Sqrt[1-8x])/(4x) + O[x]^(d+1)], x, d];
h[3][d_] := Coefficient[# (# - 1)^7(5 #^9 - 60 #^8 + 675 #^7 - 2947 #^6 + 10005 #^5 - 20235 #^4 + 28297 #^3 - 23937 #^2 + 11418 # - 1781)/(2 (# - 2)^12(# + 1)^9) &[(1-Sqrt[1-8x])/(4x) + O[x]^(d+1)],x,d];
a3[d_] := (h[3][d] + 15h[2][d/2] + 4hO[d/2,1,{4}] + hO[d/2,0,{8}] + 18hO[d/3,1,{2}] + 10hO[d/3,0,{5}] + 12hO[d/4,1,{2}] + 2hO[d/4,0,{3,2}] + 8hO[d/4,0,{4}] + 2hO[d/6,0,{1,2,1}] + 2hO[d/6,0,{2,2}] + 30hO[d/7,0,{3}] + 8hO[d/8,0,{1,2}] + 12hO[d/9,0,{1,2}] + 4hO[d/12,0,{1,2}] + 4hO[d/12,0,{1,1,1}] + 6hO[d/14,0,{1,1,1}]) / d;
Table[a3[n],{n,23}] (* Andrei Zabolotskii, Jun 24 2025, using Mednykh & Nedela's Theorem 9 *)

Terms a(13) onwards from Andrei Zabolotskii, Jun 24 2025

A118093 Numbers of rooted hypermaps on the torus with n darts (darts are semi-edges in the particular case of ordinary maps).

Original entry on oeis.org

1, 15, 165, 1611, 14805, 131307, 1138261, 9713835, 81968469, 685888171, 5702382933, 47168678571, 388580070741, 3190523226795, 26124382262613, 213415462218411, 1740019150443861, 14162920013474475, 115112250539595093, 934419385591442091, 7576722323539318101

Offset: 3

Valery A. Liskovets, Apr 13 2006

nonn

Vincenzo Librandi, Table of n, a(n) for n = 3..500
A. Mednykh and R. Nedela, Counting unrooted hypermaps on closed orientable surface, 18th Intern. Conf. Formal Power Series & Algebr. Comb., Jun 19, 2006, San Diego, California (USA).
A. Mednykh and R. Nedela, Enumeration of unrooted hypermaps of a given genus, Discr. Math., 310 (2010), 518-526. [From _N. J. A. Sloane_, Dec 19 2009]
Mednykh, A.; Nedela, R. Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), table 3
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
P. G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.

Cf. A000108, A000257, A118094, A214817, A214818.

[&+[(2^k*(4^(n-2-k)-1)*Binomial(n+k, k))/3 : k in [0..n-3]]: n in [3..25]]; // Vincenzo Librandi, Sep 16 2018

Table[Sum[2^k (4^(n - 2 - k) - 1) Binomial[n+k, k] / 3, {k, 0, n-3}], {n, 3, 25}] (* Vincenzo Librandi, Sep 16 2018 *)

a(n) = sum(k=0, n-3, 2^k*(4^(n-2-k)-1)*binomial(n+k, k))/3; \\ Michel Marcus, Dec 11 2014

seq(N) = {
  my(x='x+O('x^(N+2)), y=(1-sqrt(1-8*x))/(4*x));
  Vec((y - 1)^3/(4*(y - 2)^2*(y + 1)));
};
seq(21) \\ Gheorghe Coserea, Nov 06 2018

Conjecture: +n*(5*n-17)*a(n) -15*(n-1)*(5*n-16)*a(n-1) +12*(20*n^2-103*n+140)*a(n-2) +32*(5*n-12)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Apr 05 2018
G.f.: (1 - 7*x + 4*x^2 - (1 - 3*x)*sqrt(1 - 8*x))/(8*(1 + x)*(1 - 8*x)); equivalently, the g.f. can be rewritten as (y - 1)^3/(4*(y - 2)^2*(y + 1)), where y=G(2*x) with G the g.f. of A000108. - Gheorghe Coserea, Nov 06 2018
a(n) ~ 2^(3*n - 4) / 3 * (1 - 10/(3*sqrt(Pi*n))). - Vaclav Kotesovec, Nov 06 2018

More terms from Michel Marcus, Dec 11 2014

A214821 Number of genus 0 unsensed hypermaps with n darts.

Original entry on oeis.org

1, 3, 6, 20, 57, 240, 954, 4566, 22641, 121823, 683307, 4004055

Offset: 1

N. J. A. Sloane, Aug 01 2012

Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3

Cf. A090371, A118094, A214819, A214820, A057005; A214821-A214823, A215017, A215018.

A214823 Number of genus 2 unsensed hypermaps with n darts.

Original entry on oeis.org

0, 0, 0, 0, 4, 39, 456, 5554, 63378, 698568, 7391499, 75807708

Offset: 1

N. J. A. Sloane, Aug 01 2012

Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3

Cf. A090371, A118094, A214819, A214820, A057005; A214821-A214823, A215017, A215018.

A215017 Number of genus 3 unsensed hypermaps with n darts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 25, 678, 15867, 307880, 5180472, 78573507

Offset: 1

N. J. A. Sloane, Aug 01 2012

Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3

Cf. A090371, A118094, A214819, A214820, A057005; A214821-A214823, A215017, A215018.

A215018 Number of unsensed hypermaps with n darts and any genus.

Original entry on oeis.org

1, 3, 7, 26, 91, 490, 2785, 20434, 171579, 1671193, 18192737, 218487504

Offset: 1

N. J. A. Sloane, Aug 01 2012

Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3

Cf. A090371, A118094, A214819, A214820, A057005; A214821-A214823, A215017, A215018.

A380453 Number of dessins d'enfants D(n,g) with n edges of genus g, read by rows.

Original entry on oeis.org

1, 3, 6, 1, 20, 6, 60, 33, 4, 291, 285, 48, 1310, 2115, 708, 30, 6975, 16533, 9807, 1155, 37746, 126501, 119436, 29910, 900, 215602, 972441, 1355400, 601364, 58032, 1262874, 7451679, 14561360, 10260804, 2112300, 54990, 7611156, 57167260, 150429819, 156469887, 57017238, 4764654

Offset: 1

Paawan Jethva, Jun 22 2025

Note that Sum_{g>=0} D(n,g) gives A057005 which is the number of dessins d'enfants with n edges (as one would hope).
We get a new genus every two edges.
n=7 is the first time we have more dessins of genus 1 than genus 0.

			Triangle D(n,g) begins:
   n\g    0      1      2      3      4      ...
   1      1
   2      3
   3      6      1
   4      20     6
   5      60     33     4
   6      291    285    48
   7      1310   2115   708    30
   8      6975   16533  9807   1155
   9      37746  126501 119436 29910  900
   ...

Paawan Jethva, Exploring the Euler Characteristics of Dessins d’Enfants, 2023, page 15.
Ján Karabáš, Enumeration of actions of cyclic groups on orientable surfaces.
A. Mednykh and R. Nedela, Recent progress in enumeration of hypermaps, J. Math. Sci., New York 226, No. 5, 635-654 (2017) and Zap. Nauchn. Semin. POMI 446, 139-164 (2016), tables 2-8 and theorems 8-12. Theorem 12 has typos; the corrected formula can be inferred from Karabáš's tables.

Cf. A057005.

Columns: A090371, A118094, A214819, A214820, A356694. A321710 is the rooted version.

Rows 10-11 from Andrei Zabolotskii, Jun 28 2025

A214822 Number of genus 1 unsensed hypermaps with n darts.

Original entry on oeis.org

0, 0, 1, 6, 30, 211, 1350, 9636, 69169, 513012, 3843024, 29107494

Offset: 1

N. J. A. Sloane, Aug 01 2012

Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3

Cf. A090371, A118094, A214819, A214820, A057005; A214821-A214823, A215017, A215018.

cp's OEIS Frontend

A090371 Number of unrooted planar 2-constellations with n digons. Also number of n-edge unrooted planar Eulerian maps with bicolored faces.

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Maple

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PARI

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A214819 Number of genus 2 sensed hypermaps with n darts.

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A214820 Number of genus 3 sensed hypermaps with n darts.

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A118093 Numbers of rooted hypermaps on the torus with n darts (darts are semi-edges in the particular case of ordinary maps).

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Magma

Mathematica

PARI

PARI

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A214821 Number of genus 0 unsensed hypermaps with n darts.

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A214823 Number of genus 2 unsensed hypermaps with n darts.

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A215017 Number of genus 3 unsensed hypermaps with n darts.

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A215018 Number of unsensed hypermaps with n darts and any genus.

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A380453 Number of dessins d'enfants D(n,g) with n edges of genus g, read by rows.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A214822 Number of genus 1 unsensed hypermaps with n darts.

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