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%I #5 Aug 09 2019 07:16:44
%S 1,2,7,112,32105,2147161102,9223372004645756887,
%T 170141183460469231537996491362807709908,
%U 57896044618658097711785492504343953921871039195927143534469727707459805807105
%N Number of 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, 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 Binomial transform of A059201.
%e The a(0) = 1 through a(2) = 7 set-systems:
%e {} {} {}
%e {{1}} {{1}}
%e {{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],{1,n}]],UnsameQ@@dual[#]&]],{n,0,3}]
%Y The non-T_0 version is A058891 shifted to the left.
%Y The covering case is A059201.
%Y The version with empty edges is A326941.
%Y The unlabeled version is A326946.
%Y Cf. A003180, A316978, A319559, A319564, A319637, A326939, A326947, A326949.
%K nonn
%O 0,2
%A _Gus Wiseman_, Aug 07 2019
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