A350789 -id:A350789 - cp's OEIS frontend

A003085 Number of weakly connected digraphs with n unlabeled nodes.

1, 2, 13, 199, 9364, 1530843, 880471142, 1792473955306, 13026161682466252, 341247400399400765678, 32522568098548115377595264, 11366712907233351006127136886487, 14669074325902449468573755897547924182

Offset: 1

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 124 and 241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Keith Briggs, Table of n, a(n) for n = 1..64
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Alexander G. Ginsberg, Firing-Rate Models in Computational Neuroscience: New Applications and Methodologies, Ph. D. Dissertation, Univ. Michigan, 2023. See p. 7.
Martin Golubitsky and Yangyang Wang, Infinitesimal homeostasis in three-node input-output networks, Journal of Mathematical Biology (2020) Vol. 80, 1163-1185.
A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
X. Li, D. S. Stones, H. Wang, H. Deng, X. Liu and G. Wang, NetMODE: Network Motif Detection without Nauty, PLoS ONE 7(12): e50093. doi:10.1371/journal.pone.0050093. - From _N. J. A. Sloane_, Feb 02 2013
Eric Weisstein's World of Mathematics, Weakly Connected Digraph

Row sums of A054733.
Column sums of A350789.
The labeled case is A003027.
Cf. A000273, A003084, A035512 (strongly connected).

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000273):
seq(a(n), n = 1..13); # Peter Luschny, Nov 21 2022

Needs["Combinatorica`"]; d[n_] := GraphPolynomial[n, x, Directed] /. x -> 1; max = 13; se = Series[ Sum[a[n]*x^n/n, {n, 1, max}] - Log[1 + Sum[ d[n]*x^n, {n, 1, max}]], {x, 0, max}]; sol = SolveAlways[ se == 0, x]; Do[ A003084[n] = a[n] /. sol[[1]], {n, 1, max}]; ClearAll[a, d]; a[n_] := (1/n)*Sum[ MoebiusMu[ n/d ] * A003084[d], {d, Divisors[n]} ]; Table[ a[n], {n, 1, max}] (* Jean-François Alcover, Feb 01 2012, after formula *)
terms = 13;
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1];
d[n_] := (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]} ]; s/n!);
A003084 = CoefficientList[Log[Sum[d[n] x^n, {n, 0, terms+1}]] + O[x]^(terms + 1), x] Range[0, terms] // Rest;
a[n_] := (1/n)*Sum[MoebiusMu[n/d] * A003084[[d]], {d, Divisors[n]}];
Table[a[n], {n, 1, terms}] (* Jean-François Alcover, Aug 30 2019, after Andrew Howroyd in A003084 *)

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 A003085(n):
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<Chai Wah Wu, Jul 05 2024

a(n) = (1/n)*Sum_{d|n} mu(n/d)*A003084(d), where mu is Moebius function.

Inverse Euler transform of A000273. - Andrew Howroyd, Dec 27 2021

More terms from Vladeta Jovovic, Jan 09 2000

A054733 Triangle of number of (weakly) connected unlabeled digraphs with n nodes and k arcs (n >=2, k >= 1).

Original entry on oeis.org

1, 1, 0, 3, 4, 4, 1, 1, 0, 0, 8, 22, 37, 47, 38, 27, 13, 5, 1, 1, 0, 0, 0, 27, 108, 326, 667, 1127, 1477, 1665, 1489, 1154, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 0, 91, 582, 2432, 7694, 19646, 42148, 77305, 122953, 170315, 206982, 220768, 207301, 171008

Offset: 2

Vladeta Jovovic, Apr 21 2000

			1,1;
0,3,4,4,1,1;
0,0,8,22,37,47,38,27,13,5,1,1;
the last batch giving the numbers of connected digraphs with 4 nodes and from 1 to 12 arcs.

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

Andrew Howroyd, Table of n, a(n) for n = 2..2661 (rows 2..20)
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 75.

Cf. A000238 (leading diagonal), A003085 (row sums), A053454 (column sums), A062735 (labeled).

Cf. A052283 (not necessarily connected), A283753 (another version), A057276 (strongly connected), A350789 (transpose).

InvEulerMTS(p)={my(n=serprec(p,x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^(2*g) )) * prod(i=1, #v, my(c=v[i]); t(c)^(c-1))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i)); s/n!}
row(n)={Vecrev(polcoef(InvEulerMTS(sum(i=0, n, G(i, y)*x^i, O(x*x^n))), n)/y)}
{ for(n=2, 6, print(row(n))) } \\ Andrew Howroyd, Jan 28 2022

A053454 Number of weakly connected digraphs with n arcs.

Original entry on oeis.org

1, 1, 4, 12, 53, 237, 1306, 7537, 47913, 322253, 2297874, 17191216, 134505656, 1095715055, 9267223594, 81162609328, 734511656413, 6856030049629, 65899370570285, 651338242941020, 6611459646337423, 68842439737228261

Offset: 0

Vladeta Jovovic, Jan 12 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column sums of A054733.
Row sums of A350789.
Cf. A003085, A053418, A350752.

\\ See A054733 for G, InvEulerMTS.
seq(n)=Vec(subst(Pol(InvEulerMTS(sum(i=0, n, G(i, y+O(y^n))*x^i, O(x*x^n)))), x, 1)) \\ Andrew Howroyd, Jan 28 2022

Inverse Euler transform of A053418. - Max Alekseyev, Jan 22 2010

Extended by Max Alekseyev, Jan 22 2010

a(0)=1 prepended by Andrew Howroyd, Jan 28 2022

A350914 Triangle read by rows: T(n,k) is the number of unlabeled weakly connected oriented graphs with n arcs and k vertices, k = 1..n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 0, 0, 2, 8, 0, 0, 0, 12, 27, 0, 0, 0, 10, 68, 91, 0, 0, 0, 4, 127, 395, 350, 0, 0, 0, 0, 144, 1144, 2170, 1376, 0, 0, 0, 0, 107, 2393, 9139, 11934, 5743, 0, 0, 0, 0, 50, 3767, 28606, 67104, 64892, 24635, 0, 0, 0, 0, 12, 4500, 71583, 288331, 468702, 352286, 108968

Offset: 0

Andrew Howroyd, Jan 29 2022

			Triangle begins:
  1;
  0, 1;
  0, 0, 3;
  0, 0, 2,  8;
  0, 0, 0, 12,  27;
  0, 0, 0, 10,  68,   91;
  0, 0, 0,  4, 127,  395,  350;
  0, 0, 0,  0, 144, 1144, 2170, 1376;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A350915.
Column sums are A086345.
Cf. A350734 (transpose), A350789 (digraphs).

\\ See A350734 for G, InvEulerMTS.
T(n)={my(p=InvEulerMTS(sum(i=0, n, G(i, y+O(y^n))*x^i, O(x*x^n)))); vector(n, n, Vec(O(x^n)+polcoef(p,n-1,y)/x, -n))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

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A003085 Number of weakly connected digraphs with n unlabeled nodes.

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A054733 Triangle of number of (weakly) connected unlabeled digraphs with n nodes and k arcs (n >=2, k >= 1).

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A053454 Number of weakly connected digraphs with n arcs.

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A350914 Triangle read by rows: T(n,k) is the number of unlabeled weakly connected oriented graphs with n arcs and k vertices, k = 1..n+1.

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