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%I #10 Aug 09 2019 07:16:37
%S 2,2,8,192,63384,4294003272,18446743983526539408,
%T 340282366920938462946865774750753349904,
%U 115792089237316195423570985008687907841019819456486779364848020385134373080448
%N Number of T_0 sets of subsets of {1..n} that cover all n vertices.
%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, counted with multiplicity. 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) = 2 * A059201(n).
%F Inverse binomial transform of A326941.
%e The a(0) = 2 through a(2) = 8 sets of subsets:
%e {} {{1}} {{1},{2}}
%e {{}} {{},{1}} {{1},{1,2}}
%e {{2},{1,2}}
%e {{},{1},{2}}
%e {{},{1},{1,2}}
%e {{},{2},{1,2}}
%e {{1},{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[#]&]],{n,0,3}]
%Y The non-T_0 version is A000371.
%Y The case without empty edges is A059201.
%Y The non-covering version is A326941.
%Y The unlabeled version is A326942.
%Y The case closed under intersection is A326943.
%Y Cf. A003180, A003181, A003465, A316978, A319564, A319637, A326940, A326947.
%K nonn
%O 0,1
%A _Gus Wiseman_, Aug 07 2019
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