A140638 -id:A140638 - cp's OEIS frontend

A369145 Number of unlabeled loop-graphs with up to n vertices such that it is possible to choose a different vertex from each edge (choosable).

1, 2, 5, 12, 30, 73, 185, 467, 1207, 3147, 8329, 22245, 60071, 163462, 448277, 1236913, 3432327, 9569352, 26792706, 75288346, 212249873, 600069431, 1700826842, 4831722294, 13754016792, 39224295915, 112048279650, 320563736148, 918388655873, 2634460759783, 7566000947867

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

a(n) is the number of graphs with loops on n unlabeled vertices with every connected component having no more edges than vertices. - Andrew Howroyd, Feb 02 2024

			The a(0) = 1 through a(3) = 12 loop-graphs (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{1},{2}}    {{1},{2}}
             {{1},{1,2}}  {{1},{1,2}}
                          {{1},{2,3}}
                          {{1,2},{1,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Without the choice condition we get A000666, labeled A006125 (shifted left).
The case of a unique choice is A087803, labeled A088957.
Without loops we have A134964, labeled A133686 (covering A367869).
For exactly n edges and no loops we have A137917, labeled A137916.
The labeled version is A368927, covering A369140.
The labeled complement is A369141, covering A369142.
For exactly n edges we have A368984, labeled A333331 (maybe).
The complement for exactly n edges is A368835, labeled A368596.
The complement is counted by A369146, labeled A369141 (covering A369142).
The covering case is A369200.
The complement for exactly n edges and no loops is A369201, labeled A369143.
A000085, A100861, A111924 count set partitions into singletons or pairs.
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.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable labeled graphs, covering A367868.
A368927 counts choosable labeled loop-graphs, covering A369140.
Cf. A000088, A006649, A062740, A140638, A368924, A369194.

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],{1,2}]], Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]]],{n,0,4}]

Partial sums of A369200.

Euler transform of A369289. - Andrew Howroyd, Feb 02 2024

a(7) onwards from Andrew Howroyd, Feb 02 2024

A140636 Number of connected graphs on n unlabeled nodes that contain at least two cycles.

Original entry on oeis.org

0, 0, 0, 2, 13, 93, 809, 11005, 260793, 11715808, 1006698524, 164059824899, 50335907853919, 29003487462805642, 31397381142761123838, 63969560113225175845492, 245871831682084026518599099, 1787331725248899088890197955308, 24636021429399867655322650752269938

Offset: 1

Washington Bomfim, May 20 2008

nonn

Original name: number of unlabeled complex components with n nodes.
We can find in "The Birth of the Giant Component", p. 2, see the first link:
"As each of the random graphs evolved, the story went, never once was there more than a single 'complex' component; i.e. there never were two or more components present simultaneously that were neither trees nor unicyclic."
So a complex component is a connected graph that is neither a tree nor an unicyclic graph.

			a(4) = 2. See the two complex components with 4 nodes in the Sloane illustration.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Svante Janson, Donald E. Knuth, Tomasz Łuczak and Boris Pittel, The Birth of the Giant Component, [DOI], Rand. Struct. Alg. 4 (3) (1993) 233-358
N. J. A. Sloane, Illustration of initial terms of A001349.

The labeled version is A140638.

Cf. A001349, A000055, A001429, A005703.

a(n) = A001349(n) - A005703(n).

a(n) = A001349(n) - A000055(n) - A001429(n).

Name changed by Andrew Howroyd, Jan 16 2022

A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 15, 32, 80, 198, 528

Offset: 0

Gus Wiseman, Dec 25 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 set-systems:
  {1}  {12}  {123}    {1234}    {12345}      {123456}
             {2}{12}  {13}{23}  {14}{234}    {125}{345}
                      {3}{123}  {23}{123}    {134}{234}
                                {4}{1234}    {15}{2345}
                                {2}{13}{23}  {34}{1234}
                                {2}{3}{123}  {5}{12345}
                                {3}{13}{23}  {1}{14}{234}
                                             {12}{13}{23}
                                             {1}{23}{123}
                                             {13}{24}{34}
                                             {14}{24}{34}
                                             {3}{14}{234}
                                             {3}{23}{123}
                                             {3}{4}{1234}
                                             {4}{14}{234}

Wikipedia, Axiom of choice.

For unlabeled graphs we have A005703, connected case of A134964.
For labeled graphs we have A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
The complement without connectedness is A367903, ranks A367907.
Without connectedness we have A368095, ranks A367906,
Complement with repeats: A368097, connected case of A368411, ranks A355529.
The complement is counted by A368409, connected case of A368094.
With repeats allowed: A368412, connected case of A368098, ranks A368100.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A001055, A140637, A302545, A306005, A321194, A321405, A367902, A368414, A368422.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]}, {i,Length[p]}])],{p,Permutations[Union@@m]}]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A368186 Number of n-covers of an unlabeled n-set.

Original entry on oeis.org

1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195

Offset: 0

Gus Wiseman, Dec 19 2023

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
  {{1}}  {{1},{2}}    {{1},{2},{3}}
         {{1},{1,2}}  {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1,2},{1,3},{2,3}}
                      {{1},{2,3},{1,2,3}}
                      {{1,2},{1,3},{1,2,3}}

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

The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.

brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]

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)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n,m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q,t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g,x,x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ Andrew Howroyd, Jan 03 2024

a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024

A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

0, 1, 2, 4, 11, 25, 75, 206, 650, 2049, 6895

Offset: 0

Gus Wiseman, Dec 26 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{2},{1,2}}  {{1,2,3,3}}
                               {{1,2,3,4}}
                               {{1},{1,2,2}}
                               {{1,2},{1,2}}
                               {{1,2},{2,2}}
                               {{1,3},{2,3}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
This is the connected case of A368098, ranks A368100.
Complement set-systems: A368409, connected case of A368094, ranks A367907.
For set-systems we have A368410, connected case of A368095, ranks A367906.
The complement is A368411, connected case of A368097, ranks A355529.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A140637, A302545, A316983, A317533, A319616, A367902, A367905, A368187.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[brute /@ Select[mpm[n],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A369147 Number of unlabeled loop-graphs covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 1, 7, 52, 411, 4440, 73886, 2128608, 111533208, 10812478194, 1945437194308, 650378721118910, 404749938336301313, 470163239887698682289, 1022592854829028310302180, 4177826139658552046624979658, 32163829440870460348768017832607, 468021728889827507080865185809438918

Offset: 0

Gus Wiseman, Jan 23 2024

nonn

			The a(0) = 0 through a(3) = 7 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{1},{2},{3},{1,2}}
                         {{1},{2},{1,2},{1,3}}
                         {{1},{2},{1,3},{2,3}}
                         {{1},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3}}
                         {{1},{2},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(3) = 7 unlabeled non-choosable loop-graphs.

Without the choice condition we have A322700, labeled A322661.
The complement for exactly n edges is A368984, labeled A333331 (maybe).
The labeled complement is A369140, covering case of A368927.
The labeled version is A369142, covering case of A369141.
This is the covering case of A369146.
The complement is counted by A369200, covering case of A369145.
Without loops we have A369202, covering case of A140637.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, labeled A006125 (shifted left).
A002494 counts unlabeled covering graphs, labeled A006129.
A007716 counts non-isomorphic multiset partitions, connected A007718.
Cf. A000088, A000612, A005703, A055621, A062740, A134964, A137917, A140638, A367868, A368835, A369199.

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],{1,2}]], Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]==0&]]],{n,0,4}]

First differences of A369146.

a(n) = A322700(n) - A369200(n). - Andrew Howroyd, Feb 02 2024

a(6) onwards from Andrew Howroyd, Feb 02 2024

A368411 Number of non-isomorphic connected multiset partitions of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 2, 6, 15, 50, 148, 509, 1725, 6218

Offset: 0

Gus Wiseman, Dec 26 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 15 multiset partitions:
  {{1},{1}}  {{1},{1,1}}    {{1},{1,1,1}}      {{1},{1,1,1,1}}
             {{1},{1},{1}}  {{1,1},{1,1}}      {{1,1},{1,1,1}}
                            {{1},{1},{1,1}}    {{1},{1},{1,1,1}}
                            {{1},{2},{1,2}}    {{1},{1,1},{1,1}}
                            {{2},{2},{1,2}}    {{1},{1},{1,2,2}}
                            {{1},{1},{1},{1}}  {{1},{1,2},{2,2}}
                                               {{1},{2},{1,2,2}}
                                               {{2},{1,2},{1,2}}
                                               {{2},{1,2},{2,2}}
                                               {{2},{2},{1,2,2}}
                                               {{3},{3},{1,2,3}}
                                               {{1},{1},{1},{1,1}}
                                               {{1},{2},{2},{1,2}}
                                               {{2},{2},{2},{1,2}}
                                               {{1},{1},{1},{1},{1}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A140638, connected case of A367867.
The complement for labeled graphs is A129271, connected case of A133686.
This is the connected case of A368097.
For set-systems we have A368409, connected case of A368094, ranks A367907.
Complement set-systems: A368410, connected case of A368095, ranks A367906.
The complement is A368412, connected case of A368098, ranks A368100.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A140637, A302545, A316983, A317533, A319616, A367903, A367905, A368187.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List /@ c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[brute /@ Select[mpm[n],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]

A369201 Number of unlabeled simple graphs with n vertices and n edges such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 7, 30, 124, 507, 2036, 8216, 33515, 138557, 583040, 2503093, 10985364, 49361893, 227342301, 1073896332, 5204340846, 25874724616, 131937166616, 689653979583, 3693193801069, 20247844510508, 113564665880028, 651138092719098, 3813739129140469

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

These are graphs with n vertices and n edges having at least two cycles in the same component.

			The a(0) = 0 through a(6) = 7 simple graphs:
  .  .  .  .  .  {{12}{13}{14}{23}{24}}  {{12}{13}{14}{15}{23}{24}}
                                         {{12}{13}{14}{15}{23}{45}}
                                         {{12}{13}{14}{23}{24}{34}}
                                         {{12}{13}{14}{23}{24}{35}}
                                         {{12}{13}{14}{23}{24}{56}}
                                         {{12}{13}{14}{23}{25}{45}}
                                         {{12}{13}{14}{25}{35}{45}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(6) = 7 unlabeled non-choosable graphs with 6 vertices and 6 edges.

Without the choice condition we have A001434, covering A006649.
The labeled version without choice is A116508, covering A367863, A367862.
The complement is counted by A137917, labeled A137916.
For any number of edges we have A140637, complement A134964.
For labeled set-systems we have A368600.
The case with loops is A368835, labeled A368596.
The labeled version is A369143, covering A369144.
A006129 counts covering graphs, unlabeled A002494.
A007716 counts unlabeled multiset partitions, connected A007718.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
Cf. A000088, A000612, A014068, A053530, A133686, A140638, A368601, A369141, A369146.

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}]

a(n) = A001434(n) - A137917(n).

a(25) onwards from Andrew Howroyd, Feb 02 2024

A369202 Number of unlabeled simple graphs covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 2, 13, 95, 826, 11137, 261899, 11729360, 1006989636, 164072166301, 50336940172142, 29003653625802754, 31397431814146891910, 63969589218557753075156, 245871863137828405124380563, 1787331789281458167615190373076, 24636021675399858912682459601585276

Offset: 0

Gus Wiseman, Jan 23 2024

nonn

These are simple graphs covering n vertices such that some connected component has at least two cycles.

			Representatives of the a(4) = 2 and a(5) = 13 simple graphs:
  {12}{13}{14}{23}{24}      {12}{13}{14}{15}{23}{24}
  {12}{13}{14}{23}{24}{34}  {12}{13}{14}{15}{23}{45}
                            {12}{13}{14}{23}{24}{35}
                            {12}{13}{14}{23}{25}{45}
                            {12}{13}{14}{25}{35}{45}
                            {12}{13}{14}{15}{23}{24}{25}
                            {12}{13}{14}{15}{23}{24}{34}
                            {12}{13}{14}{15}{23}{24}{35}
                            {12}{13}{14}{23}{24}{35}{45}
                            {12}{13}{14}{15}{23}{24}{25}{34}
                            {12}{13}{14}{15}{23}{24}{35}{45}
                            {12}{13}{14}{15}{23}{24}{25}{34}{35}
                            {12}{13}{14}{15}{23}{24}{25}{34}{35}{45}

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

Without the choice condition we have A002494, labeled A006129.
The connected case is A140636.
This is the covering case of A140637, complement A134964.
The labeled version is A367868, complement A367869.
The complement is counted by A368834.
The version with loops is A369147, complement A369200.
A005703 counts unlabeled connected choosable simple graphs, labeled A129271.
A007716 counts unlabeled multiset partitions, connected A007718.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A283877 counts unlabeled set-systems, connected A300913.
Cf. A000088, A000612, A006649, A001434, A055621, A137916, A137917, A140638, A368596, A369141, A369146.

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

First differences of A140637.

a(n) = A002494(n) - A368834(n).

A370318 Number of labeled simple graphs with n vertices and the same number of edges as covered vertices, such that the edge set is connected.

Original entry on oeis.org

0, 0, 0, 1, 19, 307, 5237, 99137, 2098946, 49504458, 1291570014, 37002273654, 1156078150969, 39147186978685, 1428799530304243, 55933568895261791, 2338378885159906196, 103995520598384132516, 4903038902046860966220, 244294315694676224001852, 12827355456239840407125363

Offset: 0

Gus Wiseman, Feb 18 2024

nonn

The case of an empty edge set is excluded.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The covering case is A057500, which is also the covering case of A370317.
This is the connected case of A367862, covering A367863.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by edge count.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A001429, A006649, A061540, A116508, A323818, A367916, A368951, A369197.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Length[#]==Length[Union@@#] && Length[csm[#]]==1&]],{n,0,5}]

\\ Compare A370317; use A057500 for efficiency.
a(n)=n!*polcoef(polcoef(exp(x*y + O(x*x^n))*(-x+log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024

Binomial transform of A057500 (if the null graph is not connected).

a(n) = n!*[x^n][y^n] exp(x*y)*(-x + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Feb 19 2024

cp's OEIS Frontend

A369145 Number of unlabeled loop-graphs with up to n vertices such that it is possible to choose a different vertex from each edge (choosable).

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A140636 Number of connected graphs on n unlabeled nodes that contain at least two cycles.

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A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

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A368186 Number of n-covers of an unlabeled n-set.

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A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

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A369147 Number of unlabeled loop-graphs covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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A368411 Number of non-isomorphic connected multiset partitions of weight n contradicting a strict version of the axiom of choice.

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A369201 Number of unlabeled simple graphs with n vertices and n edges such that it is not possible to choose a different vertex from each edge (non-choosable).

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