A003085 -id:A003085 - cp's OEIS frontend

A049512 Number of quasi-initially connected digraphs on n unlabeled nodes.

1, 2, 13, 197, 9312, 1528634, 880277970

Offset: 1

Vladeta Jovovic, Goran Kilibarda

From Sean A. Irvine, Aug 01 2021: (Start)
A quasi-initially connected digraph is a digraph containing at least one vertex v such that every vertex u in the graph can either be reached from v or v can be reached from u (while obeying the directions on the edges).
There is no known formula. (End)

Sean A. Irvine, Java program (github)
V. Jovovic and G. Kilibarda, Enumeration of labeled quasi-initially connected digraphs, Discrete Math., 224 (2000), 151-163.

Cf. A049414, A003085, A035512, A003088, A051421.

a(6)-a(7) from Sean A. Irvine, Aug 01 2021

A283753 Irregular triangular array read by rows: T(n,k) is the number of non-isomorphic unlabeled weakly connected digraphs on n nodes and with k arcs.

Original entry on oeis.org

1, 1, 1, 3, 4, 4, 1, 1, 8, 22, 37, 47, 38, 27, 13, 5, 1, 1, 27, 108, 326, 667, 1127, 1477, 1665, 1489, 1154, 707, 379, 154, 61, 16, 5, 1, 1, 91, 582, 2432, 7694, 19646, 42148, 77305, 122953, 170315, 206982, 220768, 207301, 171008, 124110, 78813, 43862, 21209, 8951, 3242, 1043, 288, 76, 17, 5, 1, 1, 350, 3024, 17314, 74676, 266364, 808620, 2144407

Offset: 1

Marko Riedel, Mar 15 2017

The range for the subindex k is from n-1 to n(n-1).

Obtained from A054733 by removing leading zeros.

			First rows are:
1;
1,    1;
3,    4,   4,   1,    1;
8,   22,  37,  47,   38,   27,   13,    5,    1,   1;
27, 108, 326, 667, 1127, 1477, 1665, 1489, 1154, 707, 379, ...

E. Palmer and F. Harary, Graphical Enumeration, Academic Press, 1973.

Marko R. Riedel, Number of distinct connected digraphs
Marko Riedel, Maple code for sequences A052283, A283753.

Cf. A000273, A000238, A003085, A052283.

\\ See A054733 for G, InvEulerMTS.
row(n)={Vecrev(polcoef(InvEulerMTS(sum(i=0, n, G(i, y)*x^i, O(x*x^n))), n)/y^(n-1))}
{ for(n=1, 6, print(row(n))) } \\ Andrew Howroyd, Jan 28 2022

A361587 Triangle read by rows: T(n,k) is the number of weakly connected digraphs on n unlabeled nodes with k strongly connected components.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 4, 4, 0, 83, 56, 36, 24, 0, 5048, 2406, 1101, 542, 267, 0, 1047008, 324917, 101307, 37017, 14947, 5647, 0, 705422362, 136882286, 27757789, 7134897, 2257234, 779257, 237317, 0, 1580348371788, 183851281949, 23086772643, 3922864504, 907027520, 258909828, 78691767, 20035307

Offset: 0

Andrew Howroyd, Mar 16 2023

			Triangle begins:
  1;
  0,       1;
  0,       1,      1;
  0,       5,      4,      4;
  0,      83,     56,     36,    24;
  0,    5048,   2406,   1101,   542,   267;
  0, 1047008, 324917, 101307, 37017, 14947, 5647;
  ...

Column k=1 is A035512.
Main diagonal is A101228.
Row sums are A003085.
Cf. A350794, A361582, A361588.

\\ See PARI link in A350794 for program code.
{ my(A=A361587triang(6)); for(n=1, #A, print(A[n])) }

A054590 Number of disconnected digraphs with n unlabeled nodes.

Original entry on oeis.org

0, 1, 3, 19, 244, 10101, 1562298, 885237542, 1795141933300, 13031553571814674, 341286507770733602176, 32523592049568306757117737, 11366810480400463340177768296746, 14669108426561606778443288692015619955, 70315685953531425166863071956073529852161120

Offset: 1

Vladeta Jovovic, Apr 14 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50

The labeled case is A054593.

Cf. A000273, A000719, A003085.

from functools import lru_cache
from itertools import product
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A054590(n):
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<Chai Wah Wu, Jul 05 2024

a(n) = A000273(n) - A003085(n).

Terms a(14) and beyond from Andrew Howroyd, Apr 18 2021

A054918 Number of connected unlabeled digraphs with n nodes such that complement is also connected.

Original entry on oeis.org

1, 1, 10, 180, 9120, 1520742, 878908844, 1791588717764, 13024366540532952, 341234368845828951004, 32522226812040344643993088, 11366680383641301437820379768750, 14669062959091969068110415719779627436

Offset: 1

N. J. A. Sloane, May 24 2000

Andrew Howroyd, Table of n, a(n) for n = 1..50
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A000273, A003085.

A000273 = Cases[Import["https://oeis.org/A000273/b000273.txt", "Table"], {, }][[All, 2]];
A003085 = Cases[Import["https://oeis.org/A003085/b003085.txt", "Table"], {, }][[All, 2]];
a[n_] := 2*A003085[[n]] - A000273[[n + 1]];
Array[a, 50] (* Jean-François Alcover, Aug 31 2019 *)

from functools import lru_cache
from itertools import product, combinations
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A054918(n):
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<Chai Wah Wu, Jul 05 2024

a(n) = 2*A003085(n) - A000273(n).

More terms from Vladeta Jovovic, Jul 19 2000

A054932 Number of unlabeled connected digraphs up to complementarity.

Original entry on oeis.org

1, 1, 7, 95, 4628, 760731, 439476534, 895794710762, 6512183359880844, 170617184427498641390, 16261113406024864291983616, 5683340191820651519596089554647, 7334531479545984537334675978032833750, 35157813638509073199087893774184443496308877

Offset: 1

N. J. A. Sloane, May 24 2000

Andrew Howroyd, Table of n, a(n) for n = 1..50
V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A000273, A003085, A003086, A054931.

A054932 := proc(n)
    A003085(n)-(A000273(n)-A003086(n))/2 ;
end proc:
seq(A054932(n),n=1..13) ; # R. J. Mathar, Mar 04 2018

A000273 = Cases[Import["https://oeis.org/A000273/b000273.txt", "Table"], {, }][[All, 2]];
A003085 = Cases[Import["https://oeis.org/A003085/b003085.txt", "Table"], {, }][[All, 2]];
A003086 = Cases[Import["https://oeis.org/A003086/b003086.txt", "Table"], {, }][[All, 2]];
a[n_] := A003085[[n]] - (A000273[[n + 1]] - A003086[[n]])/2;
Array[a, 50] (* Jean-François Alcover, Sep 01 2019 *)

a(n) = A003085(n) - (A000273(n)-A003086(n))/2. - Andrew Howroyd, Sep 17 2018

Terms a(14) and beyond from Andrew Howroyd, Sep 17 2018

A056988 Number of weakly connected digraphs on n unlabeled nodes that are not strongly connected.

Original entry on oeis.org

0, 1, 8, 116, 4316, 483835, 175048780, 212125583518, 887136857205696, 13086449050056880074, 691487226135525983266460, 132437909864451274070001431956, 92821692762628588679459049647697100, 239928160288841627346208269001938306502290

Offset: 1

Eric W. Weisstein

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Connected Digraph.

Cf. A003085, A035512.

a(n) = A003085(n) - A035512(n).

a(12)-a(14) added by Andrew Howroyd, Dec 27 2021

A361367 Number of weakly 2-connected simple digraphs with n unlabeled nodes.

Original entry on oeis.org

7, 129, 7447, 1399245, 853468061, 1774125803324, 12983268697759210, 340896057593147232397, 32512334188761655225275067, 11365639780174824680535568799361, 14668665138188644335253106665956458513, 70315069858161131939222463684374769308619684

Offset: 3

Manfred Scheucher, Mar 09 2023

nonn

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Directed variant of A002218.

Cf. A000273, A003085.

\\ See links in A339645 for combinatorial species functions.
edges(v) = {2*sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]-1)}
graphsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 2^edges(p) * sMonomial(p)); s/n!}
graphsSeries(n)={sum(k=0, n, graphsCycleIndex(k)*x^k) + O(x*x^n)}
cycleIndexSeries(n)={my(g=graphsSeries(n), gc=sLog(g), gcr=sPoint(gc)); intformal(x*sSolve( sLog( gcr/(x*sv(1)) ), gcr ), sv(1)) + sSolve(subst(gc, sv(1), 0), gcr)}
{ my(N=15); Vec(-2*x^2 + OgfSeries(cycleIndexSeries(N))) } \\ Andrew Howroyd, Mar 09 2023

Terms a(7) and beyond from Andrew Howroyd, Mar 09 2023

A361368 Number of weakly connected simple planar digraphs with n unlabeled nodes.

Original entry on oeis.org

2, 13, 199, 8782, 897604

Offset: 2

Manfred Scheucher, Mar 09 2023

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Directed variant of A003094.

Cf. A000273, A003085, A361366

A361370 Number of weakly 3-connected simple digraphs with n unlabeled nodes.

Original entry on oeis.org

42, 3270, 879508

Offset: 4

Manfred Scheucher, Mar 09 2023

M. Kirchweger, M. Scheucher, and S. Szeider, SAT-Based Generation of Planar Graphs, in preparation.

Directed variant of A006290.

Cf. A000273, A003085, A361367

cp's OEIS Frontend

A049512 Number of quasi-initially connected digraphs on n unlabeled nodes.

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A283753 Irregular triangular array read by rows: T(n,k) is the number of non-isomorphic unlabeled weakly connected digraphs on n nodes and with k arcs.

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PARI

A361587 Triangle read by rows: T(n,k) is the number of weakly connected digraphs on n unlabeled nodes with k strongly connected components.

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PARI

A054590 Number of disconnected digraphs with n unlabeled nodes.

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Python

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A054918 Number of connected unlabeled digraphs with n nodes such that complement is also connected.

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Mathematica

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A054932 Number of unlabeled connected digraphs up to complementarity.

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Maple

Mathematica

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A056988 Number of weakly connected digraphs on n unlabeled nodes that are not strongly connected.

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A361367 Number of weakly 2-connected simple digraphs with n unlabeled nodes.

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PARI

Extensions

A361368 Number of weakly connected simple planar digraphs with n unlabeled nodes.

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