A059201 -id:A059201 - cp's OEIS frontend

A316980 Number of non-isomorphic strict multiset partitions of weight n.

1, 1, 3, 8, 23, 63, 197, 588, 1892, 6140, 20734, 71472, 254090, 923900, 3446572, 13149295, 51316445, 204556612, 832467052, 3455533022, 14621598811, 63023667027, 276559371189, 1234802595648, 5606647482646, 25875459311317, 121324797470067, 577692044073205

Offset: 0

Gus Wiseman, Jul 18 2018

nonn

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows (or alternatively, with no equal columns).

Also the number of non-isomorphic multiset partitions of weight n with no equivalent vertices. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

			Non-isomorphic representatives of the a(3) = 8 multiset partitions with no equivalent vertices (first column) and with no equal blocks (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(1)(1) <-> (123)
  (1)(2)(2) <-> (1)(23)
  (1)(2)(3) <-> (1)(2)(3)

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

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091.
Cf. A316974, A316978, A316979, A316981, A316983.
Cf. A049311, A059201, A319558, A319560, A319567.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(p=sum(t=1, n, subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*polcoef(exp(p-subst(p,x,x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 21 2023

Euler transform of A319557. - Gus Wiseman, Sep 23 2018

a(7)-a(10) from Gus Wiseman, Sep 23 2018

Terms a(11) and beyond from Andrew Howroyd, Jan 19 2023

A319616 Number of non-isomorphic square multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 4, 11, 27, 80, 230, 719, 2271, 7519, 25425, 88868, 317972, 1168360, 4392724, 16903393, 66463148, 266897917, 1093550522, 4568688612, 19448642187, 84308851083, 371950915996, 1669146381915, 7615141902820, 35304535554923, 166248356878549, 794832704948402, 3856672543264073, 18984761300310500

Offset: 0

Gus Wiseman, Sep 25 2018

nonn

A multiset partition or hypergraph is square if its length (number of blocks or edges) is equal to its number of vertices.

Also the number of square integer matrices with entries summing to n and no empty rows or columns, up to permutation of rows and columns.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1}, {2}}
3: {{1,1,1}}
   {{1}, {2,2}}
   {{2}, {1,2}}
   {{1}, {2},{3}}
4: {{1,1,1,1}}
   {{1}, {1,2,2}}
   {{1}, {2,2,2}}
   {{2}, {1,2,2}}
   {{1,1}, {2,2}}
   {{1,2}, {1,2}}
   {{1,2}, {2,2}}
   {{1}, {1}, {2,3}}
   {{1}, {2}, {3,3}}
   {{1}, {3}, {2,3}}
   {{1}, {2}, {3}, {4}}
Non-isomorphic representatives of the a(4) = 11 square matrices:
. [4]
.
. [1 0]   [1 0]   [0 1]   [2 0]   [1 1]   [1 1]
. [1 2]   [0 3]   [1 2]   [0 2]   [1 1]   [0 2]
.
. [1 0 0]   [1 0 0]   [1 0 0]
. [1 0 0]   [0 1 0]   [0 0 1]
. [0 1 1]   [0 0 2]   [0 1 1]
.
. [1 0 0 0]
. [0 1 0 0]
. [0 0 1 0]
. [0 0 0 1]

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

Row sums of A321615.

Cf. A000219, A007716, A007718, A056156, A059201, A316980, A316983, A318795, A319560, A319616-A319646, A300913.

(* See A318795 for M[m, n, k]. *)
T[n_, k_] := M[k, k, n] - 2 M[k, k-1, n] + M[k-1, k-1, n];
a[0] = 1; a[n_] := Sum[T[n, k], {k, 1, n}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 16}] (* Jean-François Alcover, Nov 24 2018, after Andrew Howroyd *)

\\ See A318795 for M.
a(n) = {if(n==0, 1, sum(i=1, n, M(i,i,n) - 2*M(i,i-1,n) + M(i-1,i-1,n)))} \\ Andrew Howroyd, Nov 15 2018

\\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1,n,polcoef(G(k,n,n,y),k,y) - polcoef(G(k-1,n,n,y),k,y)))} \\ Andrew Howroyd, Jan 15 2024

a(11)-a(20) from Andrew Howroyd, Nov 15 2018

a(21) onwards from Andrew Howroyd, Jan 15 2024

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

A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 67, 30997, 2147296425, 9223372036784737528, 170141183460469231731687303625772608225, 57896044618658097711785492504343953926634992332820282019728791606173188627779

Offset: 0

Gus Wiseman, Dec 05 2023

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(2) = 1 set-system is {{1},{2},{1,2}}.
The a(3) = 67 set-systems have following 21 non-isomorphic representatives:
  {{1},{2},{1,2}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{1,3},{1,2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Wikipedia, Axiom of choice.

Multisets of multisets of this type are ranked by A355529.
The version without singletons is A367769.
The version for simple graphs is A367867, covering A367868.
The version allowing empty edges is A367901.
The complement is A367902, without singletons A367770, ranks A367906.
For a unique choice (instead of none) we have A367904, ranks A367908.
These set-systems have ranks A367907.
An unlabeled version is A368094, for multiset partitions A368097.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A368409, A368413.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,3}]

a(n) + A367904(n) + A367772(n) = A058891(n+1) = 2^(2^n-1).

a(5)-a(8) from Christian Sievers, Jul 26 2024

A367902 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 2, 7, 61, 1771, 187223, 70038280, 90111497503, 397783376192189

Offset: 0

Gus Wiseman, Dec 05 2023

			The a(2) = 7 set-systems:
  {}
  {{1}}
  {{2}}
  {{1,2}}
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}

Wikipedia, Axiom of choice.

The version for simple graphs is A133686, covering A367869.
The version without singletons is A367770.
The complement allowing empty edges is A367901.
The complement is A367903, without singletons A367769, ranks A367907.
For a unique choice we have A367904, ranks A367908.
These set-systems have ranks A367906.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A370636.

Table[Length[Select[Subsets[Subsets[Range[n]]], Select[Tuples[#],UnsameQ@@#&]!={}&]],{n,0,3}]

a(n) = A370636(2^n-1). - Alois P. Heinz, Mar 09 2024

a(6)-a(8) from Christian Sievers, Jul 25 2024

A367867 Number of labeled simple graphs with n vertices contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 7, 416, 24244, 1951352, 265517333, 68652859502, 35182667175398, 36028748718835272, 73786974794973865449, 302231454853009287213496, 2475880078568912926825399800, 40564819207303268441662426947840, 1329227995784915869870199216532048487

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

In the connected case, these are just graphs with more than one cycle.

			Non-isomorphic representatives of the a(4) = 7 graphs:
  {{1,2},{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

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

The complement is A133686, connected A129271, covering A367869.
The connected case is A140638 (graphs with more than one cycle).
The covering case is A367868.
For set-systems we have A367903, ranks A367907.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A326754, A355739, A355740, A367769, A367770, A367863, A367901, A367902, A367904.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) = A006125(n) - A133686(n). - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A367863 Number of n-vertex labeled simple graphs with n edges and no isolated vertices.

Original entry on oeis.org

1, 0, 0, 1, 15, 222, 3760, 73755, 1657845, 42143500, 1197163134, 37613828070, 1295741321875, 48577055308320, 1969293264235635, 85852853154670693, 4005625283891276535, 199166987259400191480, 10513996906985414443720, 587316057411626070658200, 34612299496604684775762261

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

			Non-isomorphic representatives of the a(4) = 15 graphs:
  {{1,2},{1,3},{1,4},{2,3}}
  {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The connected case is A057500, unlabeled A001429.
The unlabeled version is A006649.
The non-covering version is A116508.
For set-systems we have A367916, ranks A367917.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A323818 counts connected set-systems, unlabeled A323819, ranks A326749.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A003465, A006126, A305000, A316983, A319559, A323817, A326754, A367769, A367901, A367902, A367903.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Length[#]==n&]],{n,0,5}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform is A367862.

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n). - Andrew Howroyd, Dec 29 2023

Terms a(8) and beyond from Andrew Howroyd, Dec 29 2023

A319559 Number of non-isomorphic T_0 set systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 16, 35, 82, 200, 517, 1373, 3867, 11216, 33910, 105950

Offset: 0

Gus Wiseman, Sep 23 2018

In a set system, two vertices are equivalent if in every block the presence of the first is equivalent to the presence of the second. The T_0 condition means that there are no equivalent vertices.

The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 7 set systems:
1:        {{1}}
2:      {{1},{2}}
3:     {{2},{1,2}}
      {{1},{2},{3}}
4:    {{1,3},{2,3}}
     {{1},{2},{1,2}}
     {{1},{3},{2,3}}
    {{1},{2},{3},{4}}
5:  {{1},{2,4},{3,4}}
    {{2},{3},{1,2,3}}
    {{2},{1,3},{2,3}}
    {{3},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1},{2},{4},{3,4}}
  {{1},{2},{3},{4},{5}}

Cf. A007716, A007718, A049311, A053419, A056156, A059201, A283877, A305854, A306006, A316980, A317757.

Cf. A319557, A319558, A319560, A319564, A319565, A319566, A319567.

a(11)-a(15) from Bert Dobbelaere, May 04 2025

A367869 Number of labeled simple graphs covering n vertices and satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 0, 1, 4, 34, 387, 5596, 97149, 1959938, 44956945, 1154208544, 32772977715, 1019467710328, 34473686833527, 1259038828370402, 49388615245426933, 2070991708598960524, 92445181295983865757, 4376733266230674345874, 219058079619119072854095, 11556990682657196214302036

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

Number of labeled n-node graphs with at most one cycle in each component and no isolated vertices. - Andrew Howroyd, Dec 30 2023

			The a(3) = 4 graphs:
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The connected case is A129271.
The non-covering case is A133686, complement A367867.
The complement is A367868, connected A140638 (unlabeled A140636).
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A355740, A367770, A367863, A367902.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Select[Tuples[#], UnsameQ@@#&]!={}&]],{n,0,5}]

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(sqrt(1/(1-t))*exp(t/2 - 3*t^2/4 - x)))} \\ Andrew Howroyd, Dec 30 2023

E.g.f.: exp(B(x) - x - 1) where B(x) is the e.g.f. of A129271. - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A367868 Number of labeled simple graphs covering n vertices and contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 7, 381, 21853, 1790135, 250562543, 66331467215, 34507857686001, 35645472109753873, 73356936892660012513, 301275024409580265134121, 2471655539736293803311467943, 40527712706903494712385171632959, 1328579255614092966328511889576785109

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

			The a(4) = 7 graphs:
  {{1,2},{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{3,4}}
  {{1,2},{1,3},{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

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

The connected case is A140638, unlabeled A140636.
The non-covering case is A367867.
The complement is A367869, connected A129271, non-covering A133686.
The version for set-systems is A367903, ranks A367907.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A367769, A367770, A367863, A367901, A367902, A367904.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Select[Tuples[#], UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) = A006129(n) - A367869(n). - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

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A367902 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice.

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A367867 Number of labeled simple graphs with n vertices contradicting a strict version of the axiom of choice.

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A367863 Number of n-vertex labeled simple graphs with n edges and no isolated vertices.

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