%I #6 Aug 15 2019 07:30:30

%S 1,1,1,1,2,2,5,7,15,26,61

%N Number of non-isomorphic set-systems of weight n whose dual is a (strict) antichain.

%C Also the number of non-isomorphic set-systems where every vertex is the unique common element of some subset of the edges, also called non-isomorphic T_1 set-systems.

%C A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}.

%C An antichain is a set of sets, none of which is a subset of any other.

%e Non-isomorphic representatives of the a(1) = 1 through a(8) = 15 multiset partitions:

%e {1} {1}{2} {1}{2}{3} {1}{2}{12} {1}{2}{3}{23} {12}{13}{23}

%e {1}{2}{3}{4} {1}{2}{3}{4}{5} {1}{2}{13}{23}

%e {1}{2}{3}{123}

%e {1}{2}{3}{4}{34}

%e {1}{2}{3}{4}{5}{6}

%e .

%e {1}{23}{24}{34} {12}{13}{24}{34}

%e {3}{12}{13}{23} {2}{13}{14}{234}

%e {1}{2}{3}{13}{23} {1}{2}{13}{24}{34}

%e {1}{2}{3}{24}{34} {1}{2}{3}{14}{234}

%e {1}{2}{3}{4}{234} {1}{2}{3}{23}{123}

%e {1}{2}{3}{4}{5}{45} {1}{2}{3}{4}{1234}

%e {1}{2}{3}{4}{5}{6}{7} {1}{2}{34}{35}{45}

%e {1}{4}{23}{24}{34}

%e {2}{3}{12}{13}{23}

%e {1}{2}{3}{4}{12}{34}

%e {1}{2}{3}{4}{24}{34}

%e {1}{2}{3}{4}{35}{45}

%e {1}{2}{3}{4}{5}{345}

%e {1}{2}{3}{4}{5}{6}{56}

%e {1}{2}{3}{4}{5}{6}{7}{8}

%Y Cf. A007716, A059523, A283877, A293993, A326961, A326965, A326974, A326976, A326977, A326979, A327012, A327018.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Aug 15 2019