cp's OEIS Frontend

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

%I A305001 #14 Nov 27 2023 16:02:01
%S A305001 1,0,1,5,87,6398,7745253,2414573042063,56130437190053518791691,
%T A305001 286386577668298410118121281898931424413687
%N A305001 Number of labeled antichains of finite sets spanning n vertices without singletons.
%C A305001 From _Gus Wiseman_, Jul 03 2019: (Start)
%C A305001 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:
%C A305001   {{3},{1,2}}
%C A305001   {{2},{1,3}}
%C A305001   {{1},{2,3}}
%C A305001   {{1},{2},{3}}
%C A305001   {{1,2},{1,3},{2,3}}
%C A305001 (End)
%e A305001 The a(3) = 5 antichains:
%e A305001   {{1,2,3}}
%e A305001   {{1,2},{1,3}}
%e A305001   {{1,2},{2,3}}
%e A305001   {{1,3},{2,3}}
%e A305001   {{1,2},{1,3},{2,3}}
%t A305001 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]]]];
%t A305001 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,5}] (* _Gus Wiseman_, Jul 03 2019 *)
%Y A305001 The binomial transform is the non-covering case A307249.
%Y A305001 The second binomial transform is A014466.
%Y A305001 Cf. A000372, A003182, A006126, A006602, A046165, A261005, A304996, A304997, A304998, A304999, A305000, A326358, A326359.
%K A305001 nonn
%O A305001 0,4
%A A305001 _Gus Wiseman_, May 23 2018
%E A305001 a(9) from A307249 - _Dmitry I. Ignatov_, Nov 27 2023