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).

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

Offset: 0

Author

Gus Wiseman, Jan 23 2024

Keywords

• nonn

Comments

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

Examples

			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}
		

Links

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

Crossrefs

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.

Programs

Mathematica

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}]

Formula

First differences of A140637.

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