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%I #10 Oct 11 2023 22:22:32
%S 1,2,5,34,1919,18660178
%N Number of unlabeled T_0 set-systems on 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. 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 Partial sums of A319637.
%F a(n) = A326949(n)/2.
%e Non-isomorphic representatives of the a(0) = 1 through a(2) = 5 set-systems:
%e {} {} {}
%e {{1}} {{1}}
%e {{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[Union[normclut/@Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&]]],{n,0,3}]
%Y The non-T_0 version is A000612.
%Y The antichain case is A245567.
%Y The covering case is A319637.
%Y The labeled version is A326940.
%Y The version with empty edges allowed is A326949.
%Y Cf. A003180, A055621, A059052, A059201, A316978, A319559, A319564, A326942.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Aug 08 2019
%E a(5) from _Max Alekseyev_, Oct 11 2023
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