A326944 - OEIS

%I #11 Aug 15 2019 15:29:04

%S 1,1,4,58,3846,2685550,151873991914,28175291154649937052

%N Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

%C The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks 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 a(n) = Sum_{k=0..n} Stirling1(n,k)*A326881(k). - _Andrew Howroyd_, Aug 14 2019

%e The a(0) = 1 through a(2) = 4 sets of subsets:

%e   {{}}  {{},{1}}  {{},{1},{2}}

%e                   {{},{1},{1,2}}

%e                   {{},{2},{1,2}}

%e                   {{},{1},{2},{1,2}}

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

%t Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

%Y The version not closed under intersection is A059201.

%Y The non-T_0 version is A326881.

%Y The version where {} is not necessarily an edge is A326943.

%Y Cf. A003181, A003465, A055621, A182507, A245567, A316978, A319564, A326906, A326939, A326941, A326945, A326947.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Aug 08 2019

%E a(5)-a(7) from _Andrew Howroyd_, Aug 14 2019