A326970 - OEIS

%I #9 Aug 12 2019 22:32:17

%S 1,1,3,43,19251

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

%C 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 Inverse binomial transform of A326968.

%e The a(3) = 43 set-systems:

%e   {123}  {1}{23}  {1}{2}{3}     {1}{2}{3}{12}

%e          {2}{13}  {12}{13}{23}  {1}{2}{3}{13}

%e          {3}{12}  {1}{23}{123}  {1}{2}{3}{23}

%e                   {2}{13}{123}  {1}{2}{13}{23}

%e                   {3}{12}{123}  {1}{2}{3}{123}

%e                                 {1}{3}{12}{23}

%e                                 {2}{3}{12}{13}

%e                                 {1}{12}{13}{23}

%e                                 {2}{12}{13}{23}

%e                                 {3}{12}{13}{23}

%e                                 {12}{13}{23}{123}

%e .

%e   {1}{2}{3}{12}{13}     {1}{2}{3}{12}{13}{23}    {1}{2}{3}{12}{13}{23}{123}

%e   {1}{2}{3}{12}{23}     {1}{2}{3}{12}{13}{123}

%e   {1}{2}{3}{13}{23}     {1}{2}{3}{12}{23}{123}

%e   {1}{2}{12}{13}{23}    {1}{2}{3}{13}{23}{123}

%e   {1}{2}{3}{12}{123}    {1}{2}{12}{13}{23}{123}

%e   {1}{2}{3}{13}{123}    {1}{3}{12}{13}{23}{123}

%e   {1}{2}{3}{23}{123}    {2}{3}{12}{13}{23}{123}

%e   {1}{3}{12}{13}{23}

%e   {2}{3}{12}{13}{23}

%e   {1}{2}{13}{23}{123}

%e   {1}{3}{12}{23}{123}

%e   {2}{3}{12}{13}{123}

%e   {1}{12}{13}{23}{123}

%e   {2}{12}{13}{23}{123}

%e   {3}{12}{13}{23}{123}

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]

%Y Covering set-systems are A003465.

%Y Covering set-systems whose dual is strict are A059201.

%Y The T_1 case is A326961.

%Y The BII-numbers of these set-systems are A326966.

%Y The non-covering case is A326968.

%Y The unlabeled version is A326973.

%Y Cf. A006126, A059052, A059523, A326950, A326960, A326965, A326969, A326971, A326974, A326975, A326978.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Aug 10 2019