A326946 - OEIS

%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