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A326970 Number of set-systems covering n vertices whose dual is a weak antichain.

%I A326970 #9 Aug 12 2019 22:32:17
%S A326970 1,1,3,43,19251
%N A326970 Number of set-systems covering n vertices whose dual is a weak antichain.
%C A326970 A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edges consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.
%F A326970 Inverse binomial transform of A326968.
%e A326970 The a(3) = 43 set-systems:
%e A326970   {123}  {1}{23}  {1}{2}{3}     {1}{2}{3}{12}
%e A326970          {2}{13}  {12}{13}{23}  {1}{2}{3}{13}
%e A326970          {3}{12}  {1}{23}{123}  {1}{2}{3}{23}
%e A326970                   {2}{13}{123}  {1}{2}{13}{23}
%e A326970                   {3}{12}{123}  {1}{2}{3}{123}
%e A326970                                 {1}{3}{12}{23}
%e A326970                                 {2}{3}{12}{13}
%e A326970                                 {1}{12}{13}{23}
%e A326970                                 {2}{12}{13}{23}
%e A326970                                 {3}{12}{13}{23}
%e A326970                                 {12}{13}{23}{123}
%e A326970 .
%e A326970   {1}{2}{3}{12}{13}     {1}{2}{3}{12}{13}{23}    {1}{2}{3}{12}{13}{23}{123}
%e A326970   {1}{2}{3}{12}{23}     {1}{2}{3}{12}{13}{123}
%e A326970   {1}{2}{3}{13}{23}     {1}{2}{3}{12}{23}{123}
%e A326970   {1}{2}{12}{13}{23}    {1}{2}{3}{13}{23}{123}
%e A326970   {1}{2}{3}{12}{123}    {1}{2}{12}{13}{23}{123}
%e A326970   {1}{2}{3}{13}{123}    {1}{3}{12}{13}{23}{123}
%e A326970   {1}{2}{3}{23}{123}    {2}{3}{12}{13}{23}{123}
%e A326970   {1}{3}{12}{13}{23}
%e A326970   {2}{3}{12}{13}{23}
%e A326970   {1}{2}{13}{23}{123}
%e A326970   {1}{3}{12}{23}{123}
%e A326970   {2}{3}{12}{13}{123}
%e A326970   {1}{12}{13}{23}{123}
%e A326970   {2}{12}{13}{23}{123}
%e A326970   {3}{12}{13}{23}{123}
%t A326970 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326970 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326970 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]
%Y A326970 Covering set-systems are A003465.
%Y A326970 Covering set-systems whose dual is strict are A059201.
%Y A326970 The T_1 case is A326961.
%Y A326970 The BII-numbers of these set-systems are A326966.
%Y A326970 The non-covering case is A326968.
%Y A326970 The unlabeled version is A326973.
%Y A326970 Cf. A006126, A059052, A059523, A326950, A326960, A326965, A326969, A326971, A326974, A326975, A326978.
%K A326970 nonn,more
%O A326970 0,3
%A A326970 _Gus Wiseman_, Aug 10 2019