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Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.
11

%I #12 Aug 15 2019 15:30:07

%S 2,2,6,70,4078,2704780,151890105214,28175292217767880450

%N Number of T_0 sets of subsets of {1..n} that cover all n vertices 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 Inverse binomial transform of A326945.

%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A326906(k). - _Andrew Howroyd_, Aug 14 2019

%e The a(0) = 2 through a(3) = 6 sets of subsets:

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

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

%e {{},{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]]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

%Y The non-T_0 version is A326906.

%Y The case without empty edges is A309615.

%Y The non-covering version is A326945.

%Y The version not closed under intersection is A326939.

%Y Cf. A003180, A003181, A003465, A059052, A059201, A245567, A316978, A319564, A319637, A326940, A326941, A326942, A326947.

%K nonn,more

%O 0,1

%A _Gus Wiseman_, Aug 08 2019

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