A319629 Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.

1, 1, 1, 1, 1, 2, 7, 9, 29, 66, 189

Offset: 0

Author

Gus Wiseman, Sep 25 2018

Keywords

• nonn
• more

Comments

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}}.

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

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(7) = 9 antichains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
4: {{1,1,1,1}}
5: {{1,1,1,1,1}}
   {{1,1},{1,2,2}}
6: {{1,1,1,1,1,1}}
   {{1,1},{1,2,2,2}}
   {{1,1,2},{1,2,2}}
   {{1,1,2},{2,2,2}}
   {{1,1,2},{2,3,3}}
   {{1,1},{1,2},{2,2}}
   {{1,2},{1,3},{2,3}}
7: {{1,1,1,1,1,1,1}}
   {{1,1},{1,2,2,2,2}}
   {{1,1,1},{1,2,2,2}}
   {{1,1,2},{1,2,2,2}}
   {{1,1,2},{2,2,2,2}}
   {{1,1,2},{2,3,3,3}}
   {{1,1},{1,2},{2,2,2}}
   {{1,1},{1,2},{2,3,3}}
   {{1,2},{1,3},{2,3,3}}
		

Crossrefs

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

Formula

Euler transform is A319644.