A319631 - OEIS

%I #6 Oct 26 2018 12:50:18

%S 1,1,2,3,5,5,13,11,25,31,54

%N Number of non-isomorphic weight-n antichains of multisets whose dual is a chain of 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.

%e Non-isomorphic representatives of the a(1) = 1 through a(5) = 5 antichains:

%e 1: {{1}}

%e 2: {{1,1}}

%e    {{1},{1}}

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

%e    {{1,2,2}}

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

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

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

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

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

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

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

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

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

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

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

%Y Cf. A000219, A006126, A007716, A059201, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Sep 25 2018