cp's OEIS Frontend

A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.

%I A326959 #12 Aug 15 2019 15:39:38
%S A326959 1,2,5,22,297,20536,16232437,1063231148918,225402337742595309857
%N A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.
%C A326959 A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge 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}}. The T_0 condition means that the dual is strict (no repeated edges).
%F A326959 Binomial transform of A309615.
%e A326959 The a(0) = 1 through a(3) = 22 set-systems:
%e A326959   {}  {}     {}           {}
%e A326959       {{1}}  {{1}}        {{1}}
%e A326959              {{2}}        {{2}}
%e A326959              {{1},{1,2}}  {{3}}
%e A326959              {{2},{1,2}}  {{1},{1,2}}
%e A326959                           {{1},{1,3}}
%e A326959                           {{2},{1,2}}
%e A326959                           {{2},{2,3}}
%e A326959                           {{3},{1,3}}
%e A326959                           {{3},{2,3}}
%e A326959                           {{1},{1,2},{1,3}}
%e A326959                           {{2},{1,2},{2,3}}
%e A326959                           {{3},{1,3},{2,3}}
%e A326959                           {{1},{1,2},{1,2,3}}
%e A326959                           {{1},{1,3},{1,2,3}}
%e A326959                           {{2},{1,2},{1,2,3}}
%e A326959                           {{2},{2,3},{1,2,3}}
%e A326959                           {{3},{1,3},{1,2,3}}
%e A326959                           {{3},{2,3},{1,2,3}}
%e A326959                           {{1},{1,2},{1,3},{1,2,3}}
%e A326959                           {{2},{1,2},{2,3},{1,2,3}}
%e A326959                           {{3},{1,3},{2,3},{1,2,3}}
%t A326959 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326959 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]
%Y A326959 The covering case is A309615.
%Y A326959 T_0 set-systems are A326940.
%Y A326959 The version with empty edges allowed is A326945.
%Y A326959 Cf. A051185, A058891, A059201, A316978, A319559, A309615, A319637, A326943, A326944, A326946, A326947, A326959.
%K A326959 nonn,more
%O A326959 0,2
%A A326959 _Gus Wiseman_, Aug 13 2019
%E A326959 a(5)-a(8) from _Andrew Howroyd_, Aug 14 2019