A090371 -id:A090371 - cp's OEIS frontend

A118094 Numbers of unrooted hypermaps on the torus with n darts up to orientation-preserving homeomorphism (darts are semi-edges in the particular case of ordinary maps).

1, 6, 33, 285, 2115, 16533, 126501, 972441, 7451679, 57167260, 438644841, 3369276867, 25905339483, 199408447446, 1536728368389, 11856420991413, 91579955286519, 708146055343668, 5481535740059577, 42473608898628639

Offset: 3

Valery A. Liskovets, Apr 13 2006

nonn

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

Cf. A090371, A118093.

Phi2 := proc(l)
    local a,k ;
    a := 0 ;
    for k in numtheory[divisors](l) do
        a := a+numtheory[mobius](l/k)*k^2 ;
    end do:
    a ;
end proc:
h0 := proc(m)
    if type(m,integer) then
        binomial(2*m,m)*3*2^(m-1)/(m+1)/(m+2) ;
    else
        0;
    end if;
end proc:
h1 := proc(n)
    local a;
    a := 0 ;
    if n >= 3 and type(n,integer) then
        a := add(2^k*(4^(n-2-k)-1)*binomial(n+k,k),k=0..n-3) ;
    end if;
    a/3 ;
end proc:
A118094 := proc(n)
    binomial(n/2+2,4)*h0(n/2) ;
    %+2*binomial(n/3+2,3)*h0(n/3) ;
    %+6*binomial(n/4+2,3)*h0(n/4) ;
    a := %+12*binomial(n/6+2,3)*h0(n/6) ;
    for l in numtheory[divisors](n) do
        if modp(n,l) = 0 then
            a := a+h1(n/l)*Phi2(l) ;
        end if;
    end do:
    a/n ;
end proc:
seq(A118094(n),n=3..14) ; # R. J. Mathar, Dec 17 2014

h0[n_] := If[Denominator[n] == 1, 3*2^(n-1)*Binomial[2*n, n]/((n+1)*(n+2)), 0]; h1[n_] := Sum[(4^(n-2-k)-1)*Binomial[n+k, k]*2^k, {k, 0, n-3}]/3; phi2[n_] := Sum[MoebiusMu[n/d]*d^2, {d, Divisors[n]}]; a[n_] := (Binomial[n/2+2, 4]*h0[n/2] +  2*Binomial[n/3+2, 3]*h0[n/3]+6*Binomial[n/4+2, 3]*h0[n/4] + 12*Binomial[n/6+2, 3]*h0[n/6] + Sum[ phi2[d]*h1[n/d], {d, Divisors[n]}])/n; Table[a[n], {n, 3, 22}] (* Jean-François Alcover, Dec 18 2014, translated from PARI *)

h0(n) = if(denominator(n)==1, 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2)), 0);
h1(n) = sum(k=0, n-3, (4^(n-2-k)-1)*binomial(n+k, k)<Michel Marcus, Dec 11 2014 ; corrected by Charles R Greathouse IV, Dec 17 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

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

A090375 Number of unrooted Eulerian maps with bicolored faces which are self-isomorphic under reversing the colors.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 40, 93, 224, 538, 1344, 3352, 8448, 21573, 54912, 143037, 366080, 968083, 2489344, 6664856, 17199104, 46515759, 120393728, 328382874, 852017152, 2340706462, 6085836800, 16822999572, 43818024960, 121777594508, 317680680960, 887053276477

Offset: 1

Valery A. Liskovets, Dec 01 2003

M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.
V. Liskovets, Some easily derivable integer sequences, J. Integer Seq., v.3 (2000), Article 00.2.2.
V. A. Liskovets, Enumerative formulas for unrooted planar maps: a pattern, Electron. J. Combin., 11:1 (2004), R88.

Cf. A052701, A069727, A090371.

A069727[n_] := (1/(2n)) (3*2^(n - 1) Binomial[2 n, n]/((n + 1) (n + 2)) + Sum[EulerPhi[n/k] d[n/k] 2^(k - 2) Binomial[2 k, k], {k, Most[Divisors[n]]}]) + q[n]; A069727[0] = 1;
q[n_?EvenQ] := 2^((n - 4)/2) Binomial[n, n/2]/(n + 2); q[n_?OddQ] := 2^((n - 1)/2) Binomial[(n - 1), (n - 1)/2]/(n + 1);
d[n_] := 4 - Mod[n, 2];
h0[n_] := 3*2^(n - 1) Binomial[2n, n]/((n + 1)(n + 2));
A090371[n_] := (h0[n] + DivisorSum[n, If[# > 1, EulerPhi[#]*Binomial[n/# + 2, 2] h0[n/#], 0] &])/n;
a[n_] := 2 A069727[n] - A090371[n];
Array[a, 32] (* Jean-François Alcover, Aug 28 2019 *)

h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
a090371(n) = (h0(n) + sumdiv(n, d, (d>1)*eulerphi(d)*binomial(n/d+2, 2)*h0(n/d)))/n;
d(n) = if (n%2, 3, 4);
q(n) = if (n%2, 2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1), 2^((n-4)/2)*binomial(n, n/2)/(n+2));
a069727(n) = if (n==0, 1, q(n) + (3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2)) + sumdiv(n, k, (k!=n)*eulerphi(n/k)*d(n/k)*2^(k-2)*binomial(2*k, k)))/(2*n));
a(n) = 2*a069727(n) - a090371(n); \\ Michel Marcus, Dec 11 2014

a(n) = 2*A069727(n) - A090371(n).

a(2k+1) = 2^k*Catalan(k) = A052701(k+1).

More terms from Michel Marcus, Dec 11 2014

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

A118094 Numbers of unrooted hypermaps on the torus with n darts up to orientation-preserving homeomorphism (darts are semi-edges in the particular case of ordinary maps).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A214819 Number of genus 2 sensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A214820 Number of genus 3 sensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A214821 Number of genus 0 unsensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

A214823 Number of genus 2 unsensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

A215017 Number of genus 3 unsensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A090375 Number of unrooted Eulerian maps with bicolored faces which are self-isomorphic under reversing the colors.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A214822 Number of genus 1 unsensed hypermaps with n darts.

Views

Author

Keywords

Links

Crossrefs