A305001 Number of labeled antichains of finite sets spanning n vertices without singletons.

1, 0, 1, 5, 87, 6398, 7745253, 2414573042063, 56130437190053518791691, 286386577668298410118121281898931424413687

Offset: 0

Author

Gus Wiseman, May 23 2018

Keywords

• nonn

Comments

From Gus Wiseman, Jul 03 2019: (Start)

Also the number of antichains covering n vertices and having empty intersection (meaning there is no vertex in common to all the edges). For example, the a(3) = 5 antichains are:

{{3},{1,2}}

{{2},{1,3}}

{{1},{2,3}}

{{1},{2},{3}}

{{1,2},{1,3},{2,3}}

(End)

Examples

			The a(3) = 5 antichains:
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}
		

Crossrefs

The binomial transform is the non-covering case A307249.

The second binomial transform is A014466.

Cf. A000372, A003182, A006126, A006602, A046165, A261005, A304996, A304997, A304998, A304999, A305000, A326358, A326359.

Programs

Mathematica

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,5}] (* Gus Wiseman, Jul 03 2019 *)

Extensions

a(9) from A307249 - Dmitry I. Ignatov, Nov 27 2023