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Number of T_0 set-systems on n vertices.
16

%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