A054733 -id:A054733 - 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

A062735 Triangular array T(n,k) giving number of weakly connected digraphs with n labeled nodes and k arcs (n >= 1, 0 <= k <= n(n-1)).

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 12, 20, 15, 6, 1, 0, 0, 0, 128, 432, 768, 920, 792, 495, 220, 66, 12, 1, 0, 0, 0, 0, 2000, 11104, 33880, 73480, 123485, 166860, 184426, 167900, 125965, 77520, 38760, 15504, 4845, 1140, 190, 20, 1, 0, 0, 0, 0, 0, 41472, 337920, 1536000, 5062080

Offset: 1

Vladeta Jovovic, Jul 12 2001

			1;
0, 2, 1;
0, 0, 12, 20,   15,    6,      1;
0, 0, 0, 128,  432,  768,    920,    792,    495,    220,     66,    12, 1;
0, 0, 0,   0, 2000, 11104, 33880,  73480, 123485, 166860, 184426, 167900, ...;
0, 0, 0,   0,    0, 41472, 337920,1536000,5062080,.. ;
0, 0, 0,   0,    0,     0, 1075648,...

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; Table 76.

Cf. A003027 (row sums), A054733 (unlabeled case), A057273 (strongly connected), A097629 (diagonal), A123554 (not necessarily connected).

nn=7;s=Sum[(1+y)^(n^2-n) x^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[Log[ s]+1,{x,0,nn}],{x,y}]//Grid  (* returns triangle indexed from n = 0, Geoffrey Critzer, Oct 07 2012 *)

row(n)={Vecrev(n!*polcoef(1 + log(sum(k=0, n, (1+y)^(k*(k-1))*x^k/k!, O(x*x^n))), n))}
{ for(n=0, 5, print(row(n))) } \\ Andrew Howroyd, Jan 11 2022

E.g.f.: 1+log( Sum_{n >= 0, k >= 0} binomial(n*(n-1), k)*x^n/n!*y^k ).

A057276 Triangle T(n,k) of number of strongly connected digraphs on n unlabeled nodes and with k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 4, 16, 22, 22, 11, 5, 1, 1, 0, 0, 0, 0, 0, 1, 7, 58, 240, 565, 928, 1065, 953, 640, 359, 150, 59, 16, 5, 1, 1, 0, 0, 0, 0, 0, 0, 1, 10, 165, 1281, 6063, 19591, 47049, 87690, 131927, 163632, 170720, 151238, 115122, 75357, 42745, 20891, 8877, 3224, 1039, 286, 76, 17, 5, 1, 1

Offset: 1

Vladeta Jovovic, Goran Kilibarda, Sep 14 2000

			Triangle begins:
  [1] 1;
  [2] 0,0,1;
  [3] 0,0,0,1,2,1,1;
  [4] 0,0,0,0,1,4,16,22,22,11,5,1,1;
  ...
The number of strongly connected digraphs on 3 unlabeled nodes is 5 = 1+2+1+1.

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)

Row sums give A035512.
Column sums give A350752.
The labeled version is A057273.
Cf. A054733, A057270, A057277, A057278, A057279, A350489.

\\ See PARI link in A350489 for program code.
{ my(A=A057276rows(5)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 13 2022

Terms a(46) and beyond from Andrew Howroyd, Jan 13 2022

A350734 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n unlabeled nodes with k arcs, n >= 1, k = 0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 2, 0, 0, 0, 8, 12, 10, 4, 0, 0, 0, 0, 27, 68, 127, 144, 107, 50, 12, 0, 0, 0, 0, 0, 91, 395, 1144, 2393, 3767, 4500, 4112, 2740, 1274, 376, 56, 0, 0, 0, 0, 0, 0, 350, 2170, 9139, 28606, 71583, 145600, 244589, 339090, 387458, 361394, 271177, 159872, 71320, 22690, 4604, 456

Offset: 1

Andrew Howroyd, Jan 13 2022

			Triangle begins:
  [1] 1;
  [2] 0, 1;
  [3] 0, 0, 3, 2;
  [4] 0, 0, 0, 8, 12, 10,   4;
  [5] 0, 0, 0, 0, 27, 68, 127, 144, 107, 50, 12;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1350 (rows 1..20)

Row sums are A086345.
Column sums are A350915.
Leading diagonal is A000238.
The labeled version is A350732.
Cf. A054733, A350733, A350750, A350914 (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)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+2*x^i)); s/n!}
row(n)={Vecrev(polcoef(InvEulerMTS(sum(i=0, n, G(i, y)*x^i, O(x*x^n))), n))}
{ for(n=1, 6, print(row(n))) }

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

A350789 Triangle read by rows: T(n,k) is the number of unlabeled weakly connected digraphs with n arcs and k vertices, k = 1..n+1.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 0, 4, 8, 0, 0, 4, 22, 27, 0, 0, 1, 37, 108, 91, 0, 0, 1, 47, 326, 582, 350, 0, 0, 0, 38, 667, 2432, 3024, 1376, 0, 0, 0, 27, 1127, 7694, 17314, 16008, 5743, 0, 0, 0, 13, 1477, 19646, 74676, 117312, 84494, 24635

Offset: 0

Andrew Howroyd, Jan 28 2022

			Triangle begins:
  1;
  0, 1;
  0, 1, 3;
  0, 0, 4,  8;
  0, 0, 4, 22,   27;
  0, 0, 1, 37,  108,   91;
  0, 0, 1, 47,  326,  582,   350;
  0, 0, 0, 38,  667, 2432,  3024,  1376;
  0, 0, 0, 27, 1127, 7694, 17314, 16008, 5743;
  ...

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

Row sums are A053454.
Column sums are A003085.
Main diagonal is A000238.
Cf. A054733 (transpose), A350450 (acyclic), A350753 (strongly connected).

\\ See A054733 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])) }

A350908 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled unisolated nodes with k arcs, k = 0..n*(n-1).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 3, 4, 4, 1, 1, 0, 0, 1, 9, 23, 37, 47, 38, 27, 13, 5, 1, 1, 0, 0, 0, 3, 34, 116, 331, 669, 1128, 1477, 1665, 1489, 1154, 707, 379, 154, 61, 16, 5, 1, 1, 0, 0, 0, 1, 15, 134, 664, 2535, 7796, 19719, 42193, 77324, 122960, 170317, 206983, 220768

Offset: 1

Andrew Howroyd, Jan 28 2022

			Triangle begins:
  [1] 0;
  [2] 0, 1, 1;
  [3] 0, 0, 3, 4,  4,  1,  1;
  [4] 0, 0, 1, 9, 23, 37, 47, 38, 27, 13, 5, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)

Row sums are A053598.
Column sums are A053418.
The labeled version is A054547.
Cf. A052283, A054733.

\\ See A054733 for G.
row(n)={Vecrev(G(n,y)-G(n-1,y), n*(n-1)+1)}
{ for(n=1, 6, print(row(n))) }

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

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A003085 Number of weakly connected digraphs with n unlabeled nodes.

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A062735 Triangular array T(n,k) giving number of weakly connected digraphs with n labeled nodes and k arcs (n >= 1, 0 <= k <= n(n-1)).

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A057276 Triangle T(n,k) of number of strongly connected digraphs on n unlabeled nodes and with k arcs, k=0..n*(n-1).

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A350734 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n unlabeled nodes with k arcs, n >= 1, k = 0..n*(n-1)/2.

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A053454 Number of weakly connected digraphs with n arcs.

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A350789 Triangle read by rows: T(n,k) is the number of unlabeled weakly connected digraphs with n arcs and k vertices, k = 1..n+1.

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A350908 Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled unisolated nodes with k arcs, k = 0..n*(n-1).

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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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