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

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

Offset: 0

Author

Gus Wiseman, Dec 07 2023

Keywords

• nonn

Examples

			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}}
		

Links

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

Crossrefs

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.

Programs

Mathematica

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

PARI

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

Formula

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

Extensions

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