A319628 - OEIS

%I #6 Sep 25 2018 20:46:26

%S 1,1,2,2,3,3,10,11,37,80,233

%N Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of (not necessarily distinct) multisets.

%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},{2,2}} is {{1},{1,2,2}}.

%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

%F Euler transform is A319641.

%e Non-isomorphic representatives of the a(1) = 1 through a(6) = 10 antichains:

%e 1: {{1}}

%e 2: {{1,1}}

%e    {{1,2}}

%e 3: {{1,1,1}}

%e    {{1,2,3}}

%e 4: {{1,1,1,1}}

%e    {{1,1,2,2}}

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

%e 5: {{1,1,1,1,1}}

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

%e    {{1,1},{1,2,2}}

%e 6: {{1,1,1,1,1,1}}

%e    {{1,1,1,2,2,2}}

%e    {{1,1,2,2,3,3}}

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

%e    {{1,1},{1,2,2,2}}

%e    {{1,1,2},{1,2,2}}

%e    {{1,1,2},{2,2,2}}

%e    {{1,1,2},{2,3,3}}

%e    {{1,1},{1,2},{2,2}}

%e    {{1,2},{1,3},{2,3}}

%Y Cf. A006126, A007716, A007718, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319558, A319565, A319616-A319646.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Sep 25 2018