A053530 -id:A053530 - cp's OEIS frontend

A001858 Number of forests of trees on n labeled nodes.

1, 1, 2, 7, 38, 291, 2932, 36961, 561948, 10026505, 205608536, 4767440679, 123373203208, 3525630110107, 110284283006640, 3748357699560961, 137557910094840848, 5421179050350334929, 228359487335194570528, 10239206473040881277575, 486909744862576654283616

Offset: 0

N. J. A. Sloane and Simon Plouffe

The number of integer lattice points in the permutation polytope of {1,2,...,n}. - Max Alekseyev, Jan 26 2010
Equals the number of score sequences for a tournament on n vertices. See Prop. 7 of the article by Bartels et al., or Example 3.1 in the article by Stanley. - David Radcliffe, Aug 02 2022
Number of labeled acyclic graphs on n vertices. The unlabeled version is A005195. The covering case is A105784, connected A000272. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of the a(4) = 38 forests:
  {}  {12}  {12,13}  {12,13,14}
      {13}  {12,14}  {12,13,24}
      {14}  {12,23}  {12,13,34}
      {23}  {12,24}  {12,14,23}
      {24}  {12,34}  {12,14,34}
      {34}  {13,14}  {12,23,24}
            {13,23}  {12,23,34}
            {13,24}  {12,24,34}
            {13,34}  {13,14,23}
            {14,23}  {13,14,24}
            {14,24}  {13,23,24}
            {14,34}  {13,23,34}
            {23,24}  {13,24,34}
            {23,34}  {14,23,24}
            {24,34}  {14,23,34}
                     {14,24,34}
(End)

B. Bollobas, Modern Graph Theory, Springer, 1998, p. 290.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..388 (first 101 terms from T. D. Noe)
J. E. Bartels, J. Mount, and D. J. A. Welsh, The polytope of win vectors. Annals of Combinatorics 1, 1-15 (1997). https://doi.org/10.1007/BF02558460
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.o
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]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 132.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 7, arXiv:1709.08416 [math.CO], 2017.
Anton Izosimov, Matrix polynomials, generalized Jacobians, and graphical zonotopes, arXiv:1506.05179 [math.AG], 2015.
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 pp. 32-33.
Arun P. Mani and Rebecca J. Stones, Congruences for weighted number of labeled forests, INTEGERS 16 (2016). #A17.
J. Pitman, Coalescent Random Forests, J. Combin. Theory, A85 (1999), 165-193.
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
J. Riordan, Letter to N. J. A. Sloane, Oct 19 1970
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1975
John Riordan and N. J. A. Sloane, Correspondence, 1974
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
R. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math 6.6 (1980): 333-342.
L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581.
E. M. Wright, A relationship between two sequences, Proc. London Math. Soc. (3) 17 (1967) 296-304.
Index entries for sequences related to forests

The connected case is A000272, rooted A000169.
The unlabeled version is A005195, connected A000055.
The covering case is A105784, unlabeled A144958.
Row sums of A138464.
For triangles instead of cycles we have A213434, covering A372168.
For a unique cycle we have A372193, covering A372195.
A002807 counts cycles in a complete graph.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A006572, A006573, A088956.
Cf. A053530, A054548, A137916, A263340, A322661, A367863, A372176.

exp(x+x^2+add(n^(n-2)*x^n/n!, n=3..50));
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*j^(j-2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 15 2008
# third Maple program:
F:= exp(-LambertW(-x)*(1+LambertW(-x)/2)):
S:= series(F,x,51):
seq(coeff(S,x,j)*j!, j=0..50); # Robert Israel, May 21 2015

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[t-t^2/2],{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2012 *)
nmax = 20; CoefficientList[Series[-LambertW[-x]/(x*E^(LambertW[-x]^2/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 19 2019 *)

a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n-1,n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!)) /* Michael Somos, Aug 22 2002 */

E.g.f.: exp( Sum_{n>=1} n^(n-2)*x^n/n! ). This implies (by a theorem of Wright) that a(n) ~ exp(1/2)*n^(n-2). - N. J. A. Sloane, May 12 2008 [Corrected by Philippe Flajolet, Aug 17 2008]
E.g.f.: exp(T - T^2/2), where T = T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is Euler's tree function (see A000169). - Len Smiley, Dec 12 2001
Shifts 1 place left under the hyperbinomial transform (cf. A088956). - Paul D. Hanna, Nov 03 2003
a(0) = 1, a(n) = Sum_{j=0..n-1} C(n-1,j) (j+1)^(j-1) a(n-1-j) if n>0. - Alois P. Heinz, Sep 15 2008

More terms from Michael Somos, Aug 22 2002

A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 15, 222, 3670, 68820, 1456875, 34506640, 906073524, 26154657270, 823808845585, 28129686128940, 1035350305641990, 40871383866109888, 1722832666898627865, 77242791668604946560, 3670690919234354407000, 184312149879830557190940, 9751080154504005703189791

Offset: 0

Washington Bomfim, Feb 22 2008

Also the number of labeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. The version without the choice condition is A116508, covering A367863. - Gus Wiseman, Jan 25 2024

			a(6) = 3670 because A057500(6) = 3660 and two triangles can be labeled in 10 ways.
From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(4) = 15 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}
                        {12,13,14,24}
                        {12,13,14,34}
                        {12,13,23,24}
                        {12,13,23,34}
                        {12,13,24,34}
                        {12,14,23,24}
                        {12,14,23,34}
                        {12,14,24,34}
                        {12,23,24,34}
                        {13,14,23,24}
                        {13,14,23,34}
                        {13,14,24,34}
                        {13,23,24,34}
                        {14,23,24,34}
(End)

V. F. Kolchin, Random Graphs. Encyclopedia of Mathematics and Its Applications 53. Cambridge Univ. Press, Cambridge, 1999.

Alois P. Heinz, Table of n, a(n) for n = 0..386 (terms n = 151..200 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

The connected case is A057500.
Row sums of A106239.
The unlabeled version is A137917.
Diagonal of A144228.
The version with loops appears to be A333331, unlabeled A368984.
Column k = 0 of A368924.
The complement is counted by A369143, unlabeled A369201, covering A369144.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable simple graphs, covering A367869.
A140637 counts unlabeled non-choosable graphs, covering A369202.
A367867 counts non-choosable graphs, covering A367868.
Cf. A001434, A006649, A053530, A116508, A129271, A134964, A140638, A367862, A367863, A369191.

cy:= proc(n) option remember;
       binomial(n-1, 2)*add((n-3)!/(n-2-t)!*n^(n-2-t), t=1..n-2)
     end:
T:= proc(n,k) option remember; `if`(k=0, 1, `if`(k<0 or n T(n,n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2008

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Exp[Log[1/(1 - t)]/2 - t/2 - t^2/4], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jan 24 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

A057500(p) = (p-1)! * p^p /2 * sum(k = 3,p, 1/(p^k*(p-k)!)); /* Vladeta Jovovic, A057500. */
F(n,N) = { my(s = 0, K, D, Mc); forpart(P = n, D = Set(P); K = vector(#D);
for(i=1, #D, K[i] = #select(x->x == D[i], Vec(P)));
Mc = n!/prod(i=1,#D, K[i]!);
s += Mc * prod(i = 1, #D, A057500(D[i])^K[i] / ( D[i]!^K[i])) , [3, n], [N, N]); s };
a(n)= {my(N); sum(N = 1, n, F(n,N))};

seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-log(1+w)/2 + w/2 - w^2/4)))} \\ Andrew Howroyd, May 18 2021

a(n) = Sum_{N = 1..n} ((n!/N!) * Sum_{n_1 + n_2 + ... + n_N = n} Product_{i = 1..N} ( A057500(n_i) / n_i! ) ). [V. F. Kolchin p. 31, (1.4.2)] replacing numerator terms n_i^(n_i-2) by A057500(n_i).
a(n) = A144228(n,n). - Alois P. Heinz, Sep 15 2008
E.g.f.: exp(B(T(x))) where B(x) = (log(1/(1-x))-x-x^2/2)/2 and T(x) is the e.g.f. for A000169 (labeled rooted trees). - Geoffrey Critzer, Jan 24 2012
a(n) ~ 2^(-1/4)*exp(-3/4)*GAMMA(3/4)*n^(n-1/4)/sqrt(Pi) * (1-7*Pi/(12*GAMMA(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Aug 16 2013
E.g.f.: exp(B(x)) where B(x) is the e.g.f. of A057500. - Andrew Howroyd, May 18 2021

a(0)=1 prepended by Andrew Howroyd, May 18 2021

A137917 a(n) is the number of unlabeled graphs on n nodes whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 14, 35, 97, 264, 733, 2034, 5728, 16101, 45595, 129327, 368093, 1049520, 2999415, 8584857, 24612114, 70652441, 203075740, 584339171, 1683151508, 4852736072, 14003298194, 40441136815, 116880901512, 338040071375, 978314772989, 2833067885748, 8208952443400

Offset: 0

Washington Bomfim, Feb 24 2008

nonn

a(n) is the number of simple unlabeled graphs on n nodes whose components have exactly one cycle. - Geoffrey Critzer, Oct 12 2012

Also the number of unlabeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 5 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}
                        {12,13,24,34}  {12,13,14,23,25}
                                       {12,13,14,23,45}
                                       {12,13,14,25,35}
                                       {12,13,24,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..500
F. Ruskey, Combinatorial Generation, p. 79, Eq.(4.27), 2003
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

Cf. A008483, A137918.
The connected case is A001429.
Without the choice condition we have A001434, covering A006649.
For any number of edges we have A134964, complement A140637.
The labeled version is A137916.
The version with loops is A369145, complement A368835.
The complement is counted by A369201, labeled A369143, covering A369144.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
Cf. A000088, A000612, A007716, A014068, A053530, A116508, A133686, A140638, A368601, A369141, A369146.

Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[DihedralGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]]x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,3,nn}]],1];CoefficientList[Series[Product[1/(1-x^i)^c[[i]],{i,1,nn-1}],{x,0,nn}],x]   (* Geoffrey Critzer, Oct 12 2012, after code given by Robert A. Russell in A000081 *)
brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])],{p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n],{2}],{n}],Select[Tuples[#],UnsameQ@@#&]!={}&]]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

a(n) = Sum_{1*j_1 + 2*j_2 + ... = n} (Product_{i=3..n} binomial(A001429(i) + j_i -1, j_i)). [F. Ruskey p. 79, (4.27) with n replaced by n+1, and a_i replaced by A001429(i)].

Euler transform of A001429. - Geoffrey Critzer, Oct 12 2012

Edited by Washington Bomfim, Jun 27 2012
Terms a(30) and beyond from Andrew Howroyd, May 05 2018
Offset changed to 0 by Gus Wiseman, Jan 27 2024

A046165 Number of minimal covers of n objects.

Original entry on oeis.org

1, 1, 2, 8, 49, 462, 6424, 129425, 3731508, 152424420, 8780782707, 710389021036, 80610570275140, 12815915627480695, 2855758994821922882, 892194474524889501292, 391202163933291014701953, 240943718535427829240708786, 208683398342300491409959279244

Offset: 0

Eric W. Weisstein

nonn

No edge of a minimal cover can be a subset of any other, so minimal covers are antichains, but the converse is not true. - Gus Wiseman, Jul 03 2019

a(n) is the number of undirected graphs on n nodes for which the intersection number and independence number are equal. See Proposition 2.3.7 and Theorem 2.3.3 of the Deligeorgaki et al. paper below. - Alex Markham, Oct 13 2022

			From _Gus Wiseman_, Jul 02 2019: (Start)
The a(1) = 1 through a(3) = 8 minimal covers:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1,2},{1,3}}
                    {{1,2},{2,3}}
                    {{1},{2},{3}}
                    {{1,3},{2,3}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..113
Damian Bursztyn, François Goasdoué, and Ioana Manolescu, Optimizing Reformulation-based Query Answering in RDF, [Research Report] RR-8646, INRIA Saclay. 2014.
D. Deligeorgaki, A. Markham, P. Misra, and L. Solus, Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks, arXiv:2210.00822 [stat.ME], 2022.
Giovanni Resta, Illustration of a(4)=49.
Eric Weisstein's World of Mathematics, Minimal Cover

Cf. A035348, A000371, A003465.
Antichain covers are A006126.
Minimal covering simple graphs are A053530.
Maximal antichains are A326358.
Row sums of A035347 or of A282575.
Cf. A000372, A003182, A006602, A261005, A305844, A307249, A326359.

a:= n-> add(add((-1)^i* binomial(k,i) *(2^k-1-i)^n, i=0..k)/k!, k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2008

Table[Sum[Sum[Binomial[n,i]StirlingS2[i,k](2^k-k-1)^(n-i),{i,k,n}],{k,2,n}]+1,{n,1,20}] (* Geoffrey Critzer, Jun 27 2013 *)

E.g.f.: Sum_{n>=0} (exp(x)-1)^n*exp(x*(2^n-n-1))/n!. - Vladeta Jovovic, May 08 2004
a(n) = Sum_{k=1..n} Sum_{i=k..n} C(n,i)*Stirling2(i,k)*(2^k - k - 1)^(n - i). - Geoffrey Critzer, Jun 27 2013
a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586694585485449... if n is even, and c = JacobiTheta2(0,1/2) = EllipticTheta[2, 0, 1/2] = 2.1289312505130275585916134025753... if n is odd. - Vaclav Kotesovec, Mar 10 2014

a(0)=1 prepended by Alois P. Heinz, Feb 18 2017

A144958 Number of unlabeled acyclic graphs covering n vertices.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 10, 17, 39, 77, 176, 381, 891, 2057, 4941, 11915, 29391, 73058, 184236, 468330, 1202349, 3108760, 8097518, 21218776, 55925742, 148146312, 394300662, 1053929982, 2828250002, 7617271738, 20584886435, 55802753243

Offset: 0

Washington Bomfim, Sep 27 2008

nonn

a(n) is the number of forests with n unlabeled nodes without isolated vertices. This follows from the fact that for n>0 A005195(n-1) counts the forests with one or more isolated nodes.

The labeled version is A105784. The connected case is A000055. This is the covering case of A005195. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of non-isomorphic representatives of the a(0) = 1 through a(5) = 4 forests:
  {}    .    {12}    {13,23}    {12,34}       {12,35,45}
                                {13,24,34}    {13,24,35,45}
                                {14,24,34}    {14,25,35,45}
                                              {15,25,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The connected case is A000055.
This is the covering case of A005195, labeled A001858.
The labeled version is A105784.
For triangles instead of cycles we have A372169, non-covering A006785.
Unique cycle: A372191 (lab A372195), non-covering A236570 (lab A372193).
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A000272, A053530, A054548, A137917, A144959, A372174.

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}]]];
cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y,{k}],And@@Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&],{k,3,Length[y]}],Min@@#==First[#]&];
Table[Length[Union[Union[brute/@Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[cyc[#]]==0&]]]],{n,0,5}] (* Gus Wiseman, Apr 29 2024 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n), v=EulerT(Vec(t - t^2/2 + subst(t,x,x^2)/2))); concat([1,0], vector(#v-1, i, v[i+1]-v[i]))} \\ Andrew Howroyd, Aug 01 2024

a(n) = A005195(n) - A005195(n-1).

Name changed and 1 prepended by Gus Wiseman, Apr 29 2024.

A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 4, 23, 199, 2313, 34015, 606407, 12712643, 306407645, 8346154699, 253476928293, 8490863621050, 310937199521774, 12356288017546937, 529516578044589407, 24339848939829286381, 1194495870124420574751, 62332449791125883072149, 3446265450868329833016605

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A369199.

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{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}}
                          {{1,2},{1,3},{2,3}}

The minimal case is A001862, without loops A053530.
This is the covering case of A066383 and A369196, cf. A369192 and A369193.
The case of equality is A368597, without loops A367863.
The version without loops is A369191.
The connected case is A369197, without loops A129271.
The unlabeled version is A370169, equality A368599, non-covering A368598.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A322661 counts covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable graphs, covering A367868.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A003465, A005703, A006649, A057500, A062740, A116508, A367862, A367916.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[Union@@#]==n&&Length[#]<=n&]],{n,0,5}]

Inverse binomial transform of A369196.

A263340 Triangle read by rows: T(n,k) is the number of graphs with n vertices containing k triangles.

Original entry on oeis.org

1, 1, 2, 3, 1, 7, 2, 1, 0, 1, 14, 7, 5, 2, 3, 1, 0, 1, 0, 0, 1, 38, 23, 28, 14, 18, 9, 7, 5, 4, 1, 4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 107, 102, 141, 117, 123, 92, 80, 63, 49, 35, 35, 23, 15, 17, 10, 4, 9, 5, 2, 3, 3, 2, 2, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

Christian Stump, Oct 15 2015

Row sums give A000088.
First column is A006785.
Row lengths are 1 + binomial(n,3). - Geoffrey Critzer, Apr 13 2017

			Triangle begins:
  1;
  1;
  2;
  3,1;
  7,2,1,0,1;
  14,7,5,2,3,1,0,1,0,0,1;
  38,23,28,14,18,9,7,5,4,1,4,1,1,1,0,0,1,0,0,0,1;
  ...

Pontus von Brömssen, Rows n = 0..10, flattened
FindStat - Combinatorial Statistic Finder, The number of triangles of a graph.
Gus Wiseman, The graphs counted under row n = 5.

Row sums are A000088, labeled A006125.
Column k = 0 is A006785 (lab A213434), covering A372169 (lab A372168).
Counting edges gives A008406 (lab A084546), covering A370167 (lab A054548).
Row lengths are A050407.
The labeled version is A372170, covering A372167.
The covering case is A372173, sums A002494, labeled A006129.
Column k = 1 is A372194 (lab A372172), covering A372174 (lab A372171).
A001858 counts acyclic graphs, unlabeled A005195.
A372176 counts labeled graphs by directed cycles, covering A372175.
Cf. A000055, A053530, A137917, A137918, A140637, A144958, A322700, A370169.

Table[Table[Count[Table[Tr[MatrixPower[AdjacencyMatrix[GraphData[{n, i}]], 3]]/6, {i, 1, NumberOfGraphs[n]}], k], {k, 0, Binomial[n, 3]}], {n, 1, 7}] (* Geoffrey Critzer, Apr 13 2017 *)

Row 7 from Geoffrey Critzer, Apr 13 2017

T(0,0)=1 prepended by Alois P. Heinz, Apr 13 2017

A372169 Number of unlabeled triangle-free graphs covering n vertices.

Original entry on oeis.org

1, 0, 1, 1, 4, 7, 24, 69, 303, 1487, 10275, 92899, 1157109, 19534822, 447074367, 13764681083, 567227701549, 31139379910949

Offset: 0

Gus Wiseman, Apr 23 2024

The labeled version is A372168.

			Non-isomorphic representatives of the a(5) = 7 graphs:
  12-35-45
  13-24-35-45
  14-25-35-45
  15-25-35-45
  12-13-24-35-45
  15-23-24-35-45
  13-14-23-24-35-45

Eric Weisstein's World of Mathematics, Triangle-Free Graph
Gus Wiseman, The a(6) = 24 unlabeled triangle-free graphs covering 6 vertices.

Dominated by A002494, labeled A006129.
Covering case of A006785, labeled A213434.
The connected case is A024607.
For all cycles (not just triangles) we have A144958, labeled A105784.
The labeled version is A372168.
For a unique triangle (labeled) we have A372171, non-covering A372172.
Column k = 0 of A372173, labeled A372167.
For a unique triangle (unlabeled) we have A372174, non-covering A372194.
A001858 counts acyclic graphs, unlabeled A005195.
A006125 counts simple graphs, unlabeled A000088.
A054548 counts covering graphs by number of edges, unlabeled A370167.
A372170 counts graphs by triangles, unlabeled A263340.
Cf. A000055, A000272, A053530, A121251, A322700, A372175, A372191.

First differences of A006785.

A372170 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k triangles, 0 <= k <= binomial(n,3).

Original entry on oeis.org

1, 1, 2, 7, 1, 41, 16, 6, 0, 1, 388, 290, 195, 70, 40, 30, 0, 10, 0, 0, 1, 5789, 6980, 6910, 4560, 3030, 2292, 1230, 780, 600, 180, 236, 60, 45, 60, 0, 0, 15, 0, 0, 0, 1, 133501, 235270, 313705, 302505, 260890, 222509, 174615, 126780, 102970, 67165, 50134, 37485, 20370, 17990, 11445, 6552, 4515, 3570, 1680, 1785, 154, 735, 455, 140, 0, 105, 105, 0, 0, 0, 21, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 23 2024

			Triangle begins:
     1
     1
     2
     7    1
    41   16    6    0    1
   388  290  195   70   40   30    0   10    0    0    1
   ...
For example, the T(4,1) = 16 graphs are:
  12-13-23
  12-14-24
  13-14-34
  23-24-34
  12-13-14-23
  12-13-14-24
  12-13-14-34
  12-13-23-24
  12-13-23-34
  12-14-23-24
  12-14-24-34
  12-23-24-34
  13-14-23-34
  13-14-24-34
  13-23-24-34
  14-23-24-34

Andrew Howroyd, Table of n, a(n) for n = 0..340 (rows 0..10)

Row sums are A006125, covering A006129.
Row lengths are A050407.
Counting edges instead of triangles gives A084546, covering A054548.
Column k = 0 is A213434, covering A372168.
The unlabeled version is A263340.
The covering case is A372167, unlabeled A372173.
Column k = 1 is A372172, covering A372171.
For all cycles (not just triangles) we have A372176, covering A372175.
A001858 counts acyclic graphs, unlabeled A005195.
A367867 counts non-choosable graphs, covering A367868.
A372193 counts unicyclic graphs, unlabeled A236570, covering A372191.
Cf. A000088, A000272, A002494, A006785, A053530, A062734, A121251, A372194.

cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}]&&MemberQ[y,{#[[1]],#[[3]]}]&&MemberQ[y,{#[[2]],#[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[cys[#]]==k&]],{n,0,5},{k,0,Binomial[n,3]}]

Binomial transform of columns of A372167.

a(42) onwards from Andrew Howroyd, Dec 29 2024

A372176 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices with exactly 2k directed cycles of length > 2.

Original entry on oeis.org

1, 1, 2, 7, 1, 38, 19, 0, 6, 0, 0, 0, 1, 291, 317, 15, 220, 0, 0, 70, 55, 0, 0, 0, 0, 30, 15, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 25 2024

A directed cycle in a simple (undirected) graph is a sequence of distinct vertices, up to rotation, such that there are edges between all consecutive elements, including the last and the first.

			Triangle begins (zeros shown as dots):
   1
   1
   2
   7 1
   38 19 . 6 ... 1
   291 317 15 220 .. 70 55 .... 30 15 ........ 10 ............... 1
The T(4,3) = 6 graphs:
  12,13,14,23,24
  12,13,14,23,34
  12,13,14,24,34
  12,13,23,24,34
  12,14,23,24,34
  13,14,23,24,34

Column k = 0 is A001858 (unlabeled A005195), covering A105784.
Row lengths are A002807 + 1.
Row sums are A006125, unlabeled A000088.
Counting edges instead of cycles gives A084546 (covering A054548), unlabeled A008406 (covering A370167).
Counting triangles instead of cycles gives A372170 (covering A372167), unlabeled A263340 (covering A372173).
The covering case is A372175.
Column k = 1 is A372193 (covering A372195), unlabeled A236570.
A006129 counts graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000272, A053530, A137916, A144958, A213434, A367863, A372168, A372169, A372171, A372172, A372191.

cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y,{k}], And@@Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&], {k,3,Length[y]}],Min@@#==First[#]&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cyc[#]]==2k&]], {n,0,4}, {k,0,Length[cyc[Subsets[Range[n],{2}]]]/2}]

cp's OEIS Frontend

A001858 Number of forests of trees on n labeled nodes.

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A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

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A137917 a(n) is the number of unlabeled graphs on n nodes whose components are unicyclic graphs.

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A046165 Number of minimal covers of n objects.

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A144958 Number of unlabeled acyclic graphs covering n vertices.

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A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

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