A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

1, 1, 2, 10, 97, 2160, 126862, 21485262, 11105374322, 18109358131513, 95465831661532570, 1660400673336788987026, 96929369602251313489896310, 19268528295096123543660356281600, 13203875101002459910158494602665950757, 31517691852305548841992346407978317698725021

Offset: 0

Author

Gus Wiseman, Jan 07 2024

Keywords

• nonn

Examples

			Non-isomorphic representatives of the a(3) = 10 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{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}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}
		

Links

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

Crossrefs

The case of graphs is A001434, labeled A116508.

Labeled version is A136556, covering A054780, binomial transform of A367916.

The case of labeled covering graphs is A367863, binomial transform A367862.

These include the set-systems ranked by A367917.

The covering case is A368186, for graphs A006649, connected A057500.

Requiring all edges to be singletons or pairs gives A368598.

A003465 counts covers with any number of edges, 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, A007716, A283877, A367901, A367902, A367903, A368599.

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 /@ Subsets[Subsets[Range[n],{1,n}],{n}]]],{n,0,4}]

PARI

a(n) = polcoef(G(n, n), n) \\ G defined in A368186. - Andrew Howroyd, Jan 11 2024

Extensions

Terms a(6) and beyond from Andrew Howroyd, Jan 11 2024