A003087 -id:A003087 - cp's OEIS frontend

A174136 Partial sums of A003087.

1, 2, 4, 10, 41, 343, 6327, 249995, 20536020, 3445474030, 1169394086932, 798731069436512, 1094825607339329108, 3006942191342795594938, 16533858553282474307626008, 181924651566316766693192987410

Offset: 0

Jonathan Vos Post, Mar 09 2010

nonn

Partial sums of number of acyclic digraphs with n unlabeled nodes; equivalently partial sums of the number of equivalence classes of n X n real (0,1)-matrices with all eigenvalues positive, up to conjugation by permutations. The subsequence of primes in this partial sum begins: 2, 41, no more through a(18).

Cf. A003087, A003024.

a(n) = SUM[i=0..n] A003087(i).

A003024 Number of acyclic digraphs (or DAGs) with n labeled nodes.

Original entry on oeis.org

1, 1, 3, 25, 543, 29281, 3781503, 1138779265, 783702329343, 1213442454842881, 4175098976430598143, 31603459396418917607425, 521939651343829405020504063, 18676600744432035186664816926721, 1439428141044398334941790719839535103, 237725265553410354992180218286376719253505

Offset: 0

N. J. A. Sloane

Also the number of n X n real (0,1)-matrices with all eigenvalues positive. - Conjectured by Eric W. Weisstein, Jul 10 2003 and proved by McKay et al. 2003, 2004
Also the number of n X n real (0,1)-matrices with permanent equal to 1, up to permutation of rows/columns, cf. A089482. - Vladeta Jovovic, Oct 28 2009
Also the number of nilpotent elements in the semigroup of binary relations on [n]. - Geoffrey Critzer, May 26 2022
From Gus Wiseman, Jan 01 2024: (Start)
Also the number of sets of n nonempty subsets of {1..n} such that there is a unique way to choose a different element from each. For example, non-isomorphic representatives of the a(3) = 25 set-systems are:
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
These set-systems have ranks A367908, subset of A367906, for multisets A368101.
The version for no ways is A368600, any length A367903, ranks A367907.
The version for at least one way is A368601, any length A367902.
(End)

			For n = 2 the three (0,1)-matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}, {{1, 1}, {0, 1}}}.

Archer, K., Gessel, I. M., Graves, C., & Liang, X. (2020). Counting acyclic and strong digraphs by descents. Discrete Mathematics, 343(11), 112041.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 310.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, Eq. (1.6.1).
R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P Stanley, Enumerative Combinatorics I, 2nd. ed., p. 322.

Alois P. Heinz, Table of n, a(n) for n = 0..77 (first 41 terms from T. D. Noe)
T. E. Allen, J. Goldsmith, and N. Mattei, Counting, Ranking, and Randomly Generating CP-nets, 2014.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
Eunice Y.-J. Chen, A. Choi, and A. Darwiche, On Pruning with the MDL Score, JMLR: Workshop and Conference Proceedings vol 52, 98-109, 2016.
S. Engstrom, Random acyclic orientations of graphs, Master's thesis written at the department of Mathematics at the Royal Institute of Technology (KTH) in Stockholm, Jan. 2013.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Tristan Deleu, Yu Luo, Changhyun Cho, Pablo Lemos, Laurence Perreault-Levasseur, Yoshua Bengio, Xi Kang, Andrea Valerio Maccio, and Yashar Hezaveh, A Data-driven Discovery of the Causal Connection between Galaxy and Black Hole Evolution, arXiv:2410.00965 [astro-ph.GA], 2024. See p. 33.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Andrea V. Macciò, and Yashar Hezaveh, Beyond Causal Discovery for Astronomy: Learning Meaningful Representations with Independent Component Analysis, arXiv:2410.14775 [astro-ph.GA], 2024. See pp. 2, 4, 10.
Zehao Jin, Mario Pasquato, Benjamin L. Davis, Tristan Deleu, Yu Luo, Changhyun Cho, Pablo Lemos, Laurence Perreault-Levasseur, Yoshua Bengio, and Xi Kang, Causal Discovery in Astrophysics: Unraveling Supermassive Black Hole and Galaxy Coevolution, Astrophys. J. (2025) Vol. 979, 212. See 4.1.
P. L. Krapivsky, Hiring Strategies, arXiv:2412.10490 [cs.GT], 2024. See p. 12.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See p. 33.
Jack Kuipers and Giusi Moffa, Uniform generation of random acyclic digraphs, arXiv preprint arXiv:1202.6590 [stat.CO], 2012. - _N. J. A. Sloane_, Sep 14 2012
Laphou Lao, Zecheng Li, Songlin Hou, Bin Xiao, Songtao Guo, and Yuanyuan Yang, A Survey of IoT Applications in Blockchain Systems: Architecture, Consensus and Traffic Modeling, ACM Computing Surveys (CSUR, 2020) Vol. 53, No. 1, Article No. 18.
Tobias Ellegaard Larsen, Claus Thorn Ekstrøm, and Anne Helby Petersen, Score-Based Causal Discovery with Temporal Background Information, arXiv:2502.06232 [stat.ME], 2025. See p. 9.
Daniel Mallia, Towards an Unsupervised Bayesian Network Pipeline for Explainable Prediction, Decision Making and Discovery, Master's Thesis, CUNY Hunter College (2023).
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, J. Integer Sequences, 7 (2004), #04.3.3.
B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, arXiv:math/0310423 [math.CO], Oct 28 2003.
A. Motzek and R. Möller, Exploiting Innocuousness in Bayesian Networks, Preprint 2015.
Yisu Peng, Y. Jiang, and P. Radivojac, Enumerating consistent subgraphs of directed acyclic graphs: an insight into biomedical ontologies, arXiv preprint arXiv:1712.09679 [cs.DS], 2017.
J. Peters, J. Mooij, D. Janzing, and B. Schölkopf, Causal Discovery with Continuous Additive Noise Models, arXiv preprint arXiv:1309.6779 [stat.ML], 2013.
R. W. Robinson, Enumeration of acyclic digraphs, Manuscript. (Annotated scanned copy)
V. I. Rodionov, On the number of labeled acyclic digraphs, Disc. Math. 105 (1-3) (1992) 319-321
I. Shpitser, T. S. Richardson, J. M. Robins and R. Evans, Parameter and Structure Learning in Nested Markov Models, arXiv preprint arXiv:1207.5058 [stat.ML], 2012.
I. Shpitser, R. J. Evans, T. S. Richardson, and J. M. Robins, Introduction to nested Markov models, Behaviormetrika, Behaviormetrika Vol. 41, No. 1, 2014, 3-39.
R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
Christian Toth, Christian Knoll, Franz Pernkopf, and Robert Peharz, Rao-Blackwellising Bayesian Causal Inference, arXiv:2402.14781 [cs.LG], 2024.
Sumanth Varambally, Yi-An Ma, and Rose Yu, Discovering Mixtures of Structural Causal Models from Time Series Data, arXiv:2310.06312 [cs.LG], 2023.
S. Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).
Daniel Waxman, Kurt Butler, and Petar M. Djuric, Dagma-DCE: Interpretable, Non-Parametric Differentiable Causal Discovery, arXiv:2401.02930 [cs.LG], 2024.
Eric Weisstein's World of Mathematics, (0,1)-Matrix
Eric Weisstein's World of Mathematics, Acyclic Digraph
Eric Weisstein's World of Mathematics, Positive Eigenvalued Matrix
Eric Weisstein's World of Mathematics, Weisstein's Conjecture
Jun Wu and Mathias Drton, Partial Homoscedasticity in Causal Discovery with Linear Models, arXiv:2308.08959 [math.ST], 2023.
Chris Ying, Enumerating Unique Computational Graphs via an Iterative Graph Invariant, arXiv:1902.06192 [cs.DM], 2019.
Index entries for sequences related to binary matrices

Cf. A086510, A081064 (refined by # arcs), A307049 (by # descents).
Cf. A055165, which counts nonsingular {0, 1} matrices and A085656, which counts positive definite {0, 1} matrices.
Cf. A188457, A135079, A137435 (acyclic 3-multidigraphs), A188490.
For a unique sink we have A003025.
The unlabeled version is A003087.
These are the reverse-alternating sums of rows of A046860.
The weakly connected case is A082402.
A reciprocal version is A334282.
Row sums of A361718.
Cf. A003465, A058877, A058891, A059201, A088957, A133686, A323818.

p:=evalf(solve(sum((-1)^n*x^n/(n!*2^(n*(n-1)/2)), n=0..infinity) = 0, x), 50); M:=evalf(sum((-1)^(n+1)*p^n/((n-1)!*2^(n*(n-1)/2)), n=1..infinity), 40); # program for evaluation of constants p and M in the asymptotic formula, Vaclav Kotesovec, Dec 09 2013

a[0] = a[1] = 1; a[n_] := a[n] = Sum[ -(-1)^k * Binomial[n, k] * 2^(k*(n-k)) * a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 13}](* Jean-François Alcover, May 21 2012, after PARI *)
Table[2^(n*(n-1)/2)*n! * SeriesCoefficient[1/Sum[(-1)^k*x^k/k!/2^(k*(k-1)/2),{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 19 2015 *)
Table[Length[Select[Subsets[Subsets[Range[n]],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]==1&]],{n,0,5}] (* Gus Wiseman, Jan 01 2024 *)

a(n)=if(n<1,n==0,sum(k=1,n,-(-1)^k*binomial(n,k)*2^(k*(n-k))*a(n-k)))

{a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+2^k*x+x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Oct 17 2009

a(0) = 1; for n > 0, a(n) = Sum_{k=1..n} (-1)^(k+1)*C(n, k)*2^(k*(n-k))*a(n-k).
1 = Sum_{n>=0} a(n)*exp(-2^n*x)*x^n/n!. - Vladeta Jovovic, Jun 05 2005
a(n) = Sum_{k=1..n} (-1)^(n-k)*A046860(n,k) = Sum_{k=1..n} (-1)^(n-k)*k!*A058843(n,k). - Vladeta Jovovic, Jun 20 2008
1 = Sum_{n=>0} a(n)*x^n/(1 + 2^n*x)^(n+1). - Paul D. Hanna, Oct 17 2009
1 = Sum_{n>=0} a(n)*C(n+m-1,n)*x^n/(1 + 2^n*x)^(n+m) for m>=1. - Paul D. Hanna, Apr 01 2011
log(1+x) = Sum_{n>=1} a(n)*(x^n/n)/(1 + 2^n*x)^n. - Paul D. Hanna, Apr 01 2011
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is 1/E(-x) = Sum_{n >= 0} a(n)*x^n/(n!*2^C(n,2)) = 1 + x + 3*x^2/(2!*2) + 25*x^3/(3!*2^3) + 543*x^4/(4!*2^6) + ... (Stanley). Cf. A188457. - Peter Bala, Apr 01 2013
a(n) ~ n!*2^(n*(n-1)/2)/(M*p^n), where p = 1.488078545599710294656246... is the root of the equation Sum_{n>=0} (-1)^n*p^n/(n!*2^(n*(n-1)/2)) = 0, and M = Sum_{n>=1} (-1)^(n+1)*p^n/((n-1)!*2^(n*(n-1)/2)) = 0.57436237330931147691667... Both references to the article "Acyclic digraphs and eigenvalues of (0,1)-matrices" give the wrong value M=0.474! - Vaclav Kotesovec, Dec 09 2013 [Response from N. J. A. Sloane, Dec 11 2013: The value 0.474 has a typo, it should have been 0.574. The value was taken from Stanley's 1973 paper.]
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 10*x^3 + 146*x^4 + 6010*x^5 + ... appears to have integer coefficients (cf. A188490). - Peter Bala, Jan 14 2016

A000595 Number of binary relations on n unlabeled points.

Original entry on oeis.org

1, 2, 10, 104, 3044, 291968, 96928992, 112282908928, 458297100061728, 6666621572153927936, 349390545493499839161856, 66603421985078180758538636288, 46557456482586989066031126651104256, 120168591267113007604119117625289606148096, 1152050155760474157553893461743236772303142428672

Offset: 0

N. J. A. Sloane

Number of orbits under the action of permutation group S(n) on n X n {0,1} matrices. The action is defined by f.M(i,j)=M(f(i),f(j)).

Equivalently, the number of digraphs on n unlabeled nodes with loops allowed but no more than one arc with the same start and end node. - Andrew Howroyd, Oct 22 2017

			From _Gus Wiseman_, Jun 17 2019: (Start)
Non-isomorphic representatives of the a(2) = 10 relations:
  {}
  {1->1}
  {1->2}
  {1->1, 1->2}
  {1->1, 2->1}
  {1->1, 2->2}
  {1->2, 2->1}
  {1->1, 1->2, 2->1}
  {1->1, 1->2, 2->2}
  {1->1, 1->2, 2->1, 2->2}
(End)

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 76 (2.2.30)
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, pp. 14-22.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-François Alcover, Table of n, a(n) for n = 0..50 (a(0)-a(37) from Charles R. Greathouse IV)
Edward A. Bender and E. Rodney Canfield, Enumeration of connected invariant graphs, Journal of Combinatorial Theory, Series B 34.3 (1983): 268-278. See p. 274.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Alberto Casagrande, Carla Piazza, and Alberto Policriti, Is hyper-extensionality preservable under deletions of graph elements?, Preprint 2015.
Alexander Clow, Alfie Davies, and Neil Anderson McKay, Constructing All Birthday 3 Games as Digraphs, arXiv:2505.06206 [math.CO], 2025. See p. 13.
Matthew Dabkowski, N. Fan, and R. Breiger, Exploratory blockmodeling for one-mode, unsigned, deterministic networks using integer programming and structural equivalence, Social Networks, Volume 47, October 2016, Pages 93-106.
Robert L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.
Thomas M. A. Fink, Emmanuel Barillot, and Sebastian E. Ahnert, Dynamics of network motifs, 2006.
Frank Harary, Edgar M. Palmer, Robert W. Robinson, and Allen J. Schwenk, Enumeration of graphs with signed points and lines, J. Graph Theory 1 (1977), no. 4, 295-308.
Sergiy Kozerenko, On the abstract properties of Markov graphs for maps on trees, Mathematical Bilten 41:2 (2017), pp. 5-21.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Samuel Reid, On Generalizing a Temporal Formalism for Game Theory to the Asymptotic Combinatorics of S5 Modal Frames, arXiv preprint arXiv:1305.0064 [math.LO], 2013.
Marko Riedel, Counting nonisomorphic binary relations (includes Maple code).
R. W. Robinson, Notes - "A Present for Neil Sloane"
R. W. Robinson, Notes - computer printout
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
Leopold Travis, Graphical Enumeration: A Species-Theoretic Approach, arXiv:math/9811127 [math.CO], 1998.
Gus Wiseman, Non-isomorphic representatives of the a(3) = 104 digraphs.
Index entries for sequences related to binary matrices

Cf. A000088, A000273, A001173, A001174, A002416, A003087, A003216, A002724.

NSeq := function ( n ) return Sum(List(ConjugacyClasses(SymmetricGroup(n)), c -> (2^Length(Orbits(Group(Representative(c)), CartesianProduct([1..n],[1..n]), OnTuples))) * Size(c)))/Factorial(n); end; # Dan Hoey, May 04 2001

Join[{1,2}, Table[CycleIndex[Join[PairGroup[SymmetricGroup[n],Ordered], Permutations[Range[n^2-n+1,n^2]],2],s] /. Table[s[i]->2, {i,1,n^2-n}], {n,2,7}]] (* Geoffrey Critzer, Nov 02 2011 *)
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];
a[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 08 2018, after Andrew Howroyd *)
dinorm[m_]:=If[m=={},{},If[Union@@m!=Range[Max@@Flatten[m]],dinorm[m/.Apply[Rule,Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}],{1}]],First[Sort[dinorm[m,1]]]]];
dinorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#1>=aft&]}]},Union@@(dinorm[#1,aft+1]&)/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0}],{par,First/@Position[mx,Max[mx]]}]]]];
Table[Length[Union[dinorm/@Subsets[Tuples[Range[n],2]]]],{n,0,3}] (* Gus Wiseman, Jun 17 2019 *)

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) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i],v[j]))) + sum(i=1, #v, v[i])}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017

from itertools import product
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A000595(n): return int(sum(Fraction(1<Chai Wah Wu, Jul 02 2024

a(n) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...] / (1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = 2^sum {i, j>=1} (gcd(i, j)*s_i*s_j). - Christian G. Bower, Jan 05 2004

a(n) ~ 2^(n^2)/n! [McIlroy, 1955]. - Vaclav Kotesovec, Dec 19 2016

More terms from Vladeta Jovovic, Feb 07 2000

Still more terms from Dan Hoey, May 04 2001

A003025 Number of n-node labeled acyclic digraphs with 1 out-point.

Original entry on oeis.org

1, 2, 15, 316, 16885, 2174586, 654313415, 450179768312, 696979588034313, 2398044825254021110, 18151895792052235541515, 299782788128536523836784628, 10727139906233315197412684689421

Offset: 1

N. J. A. Sloane

nonn

From Gus Wiseman, Jan 02 2024: (Start)
Also the number of n-element sets of finite nonempty subsets of {1..n}, including a unique singleton, such that there is exactly one way to choose a different element from each. For example, the a(0) = 0 through a(3) = 15 set-systems are:
  .  {{1}}  {{1},{1,2}}  {{1},{1,2},{1,3}}
            {{2},{1,2}}  {{1},{1,2},{2,3}}
                         {{1},{1,3},{2,3}}
                         {{2},{1,2},{1,3}}
                         {{2},{1,2},{2,3}}
                         {{2},{1,3},{2,3}}
                         {{3},{1,2},{1,3}}
                         {{3},{1,2},{2,3}}
                         {{3},{1,3},{2,3}}
                         {{1},{1,2},{1,2,3}}
                         {{1},{1,3},{1,2,3}}
                         {{2},{1,2},{1,2,3}}
                         {{2},{2,3},{1,2,3}}
                         {{3},{1,3},{1,2,3}}
                         {{3},{2,3},{1,2,3}}
These set-systems are all connected.
The case of labeled graphs is A000169.
(End)

			a(2) = 2: o-->--o (2 ways)
a(3) = 15: o-->--o-->--o (6 ways) and
o ... o o-->--o
.\ . / . \ . /
. v v ... v v
.. o ..... o
(3 ways) (6 ways)

R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..50
Marcel et al., Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?, MathOverflow, 2021.

A diagonal of A058876.
Row sums of A350487.
The unlabeled version is A350415.
Column k=1 of A361718.
Cf. A003026, A003028, A345258.
For any number of sinks we have A003024, unlabeled A003087.
For n-1 sinks we have A058877.
For a fixed sink we have A134531 (up to sign), column k=1 of A368602.
Cf. A000169, A058891, A059201, A082402, A088957, A323818, A334282, A367904, A368600, A368601.

Table[Length[Select[Subsets[Subsets[Range[n]],{n}],Count[#,{}]==1&&Length[Select[Tuples[#],UnsameQ@@#&]]==1&]],{n,0,4}] (* _Gus Wiseman, Jan 02 2024 *)

\\ requires A058876.
my(T=A058876(20)); vector(#T, n, T[n][1]) \\ Andrew Howroyd, Dec 27 2021

\\ see Marcel et al. link (formula for a').
seq(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, a[n]=sum(k=1, n-1, (-1)^(k-1) * binomial(n,k) * (2^(n-k)-1)^k * a[n-k])); a} \\ Andrew Howroyd, Jan 01 2022

a(n) = (-1)^(n-1) * n * A134531(n). - Gus Wiseman, Jan 02 2024

More terms from Vladeta Jovovic, Apr 10 2001

A122078 Triangle read by rows: T(n,k) is the number of unlabeled acyclic digraphs with n >= 0 nodes and n-k outnodes (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 11, 16, 0, 1, 4, 25, 108, 164, 0, 1, 5, 47, 422, 2168, 3341, 0, 1, 6, 78, 1251, 15484, 88747, 138101, 0, 1, 7, 120, 3124, 79836, 1215783, 7409117, 11578037, 0, 1, 8, 174, 6925, 333004, 11620961, 199203464, 1252610909, 1961162564, 0

Offset: 0

N. J. A. Sloane, Oct 18 2006

			Triangle T(n,k) begins:
  1:
  1, 0;
  1, 1,  0;
  1, 2,  3,    0;
  1, 3, 11,   16,     0;
  1, 4, 25,  108,   164,     0;
  1, 5, 47,  422,  2168,  3341,      0;
  1, 6, 78, 1251, 15484, 88747, 138101, 0;
  ...

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Andrew Howroyd, Table of n, a(n) for n = 0..495 (rows 0..30; rows 0..15 from R. W. Robinson)
Andrew Howroyd, PARI program, Dec 2021, updated Jan 2022.

Row sums give A003087.
Diagonals include A000007, A350415.
Cf. A058876 (labeled case), A350447, A350448, A350449, A350450.

\\ See link for program code.
{ my(T=AcyclicDigraphsByNonSources(8)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 31 2021

Zero terms inserted by Andrew Howroyd, Dec 29 2021

A326225 Number of Hamiltonian unlabeled n-vertex digraphs (without loops).

Original entry on oeis.org

0, 1, 1, 4, 61, 3725, 844141, 626078904

Offset: 0

Gus Wiseman, Jun 15 2019

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

			Non-isomorphic representatives of the a(3) = 4 digraph edge-sets:
  {12,23,31}
  {12,13,21,32}
  {12,13,21,23,31}
  {12,13,21,23,31,32}

The labeled case is A326219.
The case with loops is A326226.
The undirected case is A003216.
Non-Hamiltonian unlabeled digraphs (without loops) are A326222.
Cf. A000595, A002416, A003087, A003216, A053763, A246446, A326204, A326218, A326223.

a(5)-a(7) from Sean A. Irvine, Jun 16 2019

A326226 Number of unlabeled n-vertex Hamiltonian digraphs (with loops).

Original entry on oeis.org

0, 2, 3, 24, 858

Offset: 0

Gus Wiseman, Jun 14 2019

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

			Non-isomorphic representatives of the a(2) = 3 digraph edge-sets:
  {12,21}
  {11,12,21}
  {11,12,21,22}

Wikipedia, Hamiltonian path
Gus Wiseman, Non-isomorphic representatives of the a(3) = 24 Hamiltonian digraphs

The labeled case is A326204.
The case without loops is A326225.
The undirected case is A003216 (without loops) or A326215 (with loops).
Unlabeled non-Hamiltonian digraphs are A326223.
Unlabeled digraphs with a Hamiltonian path are A326221.
Cf. A000595, A002416, A003087, A057864, A246446, A326208.

dinorm[m_]:=If[m=={},{},If[Union@@m!=Range[Max@@Flatten[m]],dinorm[m/. Apply[Rule,Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}],{1}]],First[Sort[dinorm[m,1]]]]];
dinorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#1>=aft&]}]},Union@@(dinorm[#1,aft+1]&)/@Union[Table[Map[Sort,m/. {par+aft-1->aft,aft->par+aft-1},{0}],{par,First/@Position[mx,Max[mx]]}]]]];
Table[Length[Select[Union[dinorm/@Subsets[Tuples[Range[n],2]]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 867 which is incorrect *)

A350415 Number of acyclic digraphs on n unlabeled nodes with a global source (or sink).

Original entry on oeis.org

1, 1, 3, 16, 164, 3341, 138101, 11578037, 1961162564, 668678055847, 457751797355605, 628137837068751147, 1726130748679532455689, 9493834992383031007906911, 104476428350838383854529661007, 2299979227717819421763629684068904

Offset: 1

Andrew Howroyd, Dec 29 2021

nonn

A local source (also called an out-node) is a node whose in-degree is zero. In the case of an acyclic digraph with only one local source, the source is also a global source.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Marcel et al., Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?, MathOverflow, 2021.

The labeled case is A003025.
Row sums of A350488.
A diagonal of A122078.
Cf. A003087, A345258, A350360.

A350415seq(16) \\ See PARI link in A122078 for program code.

A326221 Number of unlabeled n-vertex digraphs (with loops) containing a Hamiltonian path.

Original entry on oeis.org

0, 0, 7, 74, 2395

Offset: 0

Gus Wiseman, Jun 16 2019

A directed path is Hamiltonian if it passes through every vertex exactly once.

Wikipedia, Hamiltonian path
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.
Gus Wiseman, Non-isomorphic representatives of the a(3) = 74 digraphs containing a Hamiltonian path.

The labeled case is A326214.
The undirected case is A057864 (without loops).
Unlabeled digraphs not containing a Hamiltonian path are A326224.
Unlabeled digraphs containing a Hamiltonian cycle are A326226.
Cf. A000595, A002416, A003087, A003216, A283420.

A000595(n) = a(n) + A326224(n).

A326224 Number of unlabeled n-vertex digraphs (with loops) not containing a Hamiltonian path.

Original entry on oeis.org

1, 2, 3, 30, 649

Offset: 0

Gus Wiseman, Jun 16 2019

A directed path is Hamiltonian if it passes through every vertex exactly once.

Wikipedia, Hamiltonian path
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.
Gus Wiseman, Non-isomorphic representatives of the a(3) = 30 digraphs not containing a Hamiltonian path.

The labeled case is A326213.
The undirected case is A283420 (without loops).
Unlabeled digraphs containing a Hamiltonian path are A326221.
Unlabeled digraphs not containing a Hamiltonian cycle are A326223.
Cf. A000595, A002416, A003087, A003216, A057864.

A000595(n) = a(n) + A326221(n).

cp's OEIS Frontend

A174136 Partial sums of A003087.

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A003024 Number of acyclic digraphs (or DAGs) with n labeled nodes.

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A000595 Number of binary relations on n unlabeled points.

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A003025 Number of n-node labeled acyclic digraphs with 1 out-point.

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A122078 Triangle read by rows: T(n,k) is the number of unlabeled acyclic digraphs with n >= 0 nodes and n-k outnodes (0 <= k <= n).

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A326225 Number of Hamiltonian unlabeled n-vertex digraphs (without loops).

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A326226 Number of unlabeled n-vertex Hamiltonian digraphs (with loops).

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