A319638 Number of non-isomorphic weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.

1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 7

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(10) = 7 antichains:
1:  {{1}}
2:  {{1},{2}}
3:  {{1},{2},{3}}
4:  {{1},{2},{3},{4}}
5:  {{1},{2},{3},{4},{5}}
6:  {{1,2},{1,3},{2,3}}
    {{1},{2},{3},{4},{5},{6}}
7:  {{1},{2,3},{2,4},{3,4}}
    {{1},{2},{3},{4},{5},{6},{7}}
8:  {{1,2},{1,3},{2,4},{3,4}}
    {{1},{2},{3,4},{3,5},{4,5}}
    {{1},{2},{3},{4},{5},{6},{7},{8}}
9:  {{1,2},{1,3},{1,4},{2,3,4}}
    {{1},{2,3},{2,4},{3,5},{4,5}}
    {{1},{2},{3},{4,5},{4,6},{5,6}}
    {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
10: {{1,3},{2,4},{1,2,5},{3,4,5}}
    {{1},{2,3},{2,4},{2,5},{3,4,5}}
    {{1,2},{1,3},{2,4},{3,5},{4,5}}
    {{1,3},{1,4},{2,3},{2,4},{3,4}}
    {{1},{2},{3,4},{3,5},{4,6},{5,6}}
    {{1},{2},{3},{4},{5,6},{5,7},{6,7}}
    {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
		

Crossrefs

Cf. A006126, A007716, A049311, A059201, A283877, A293606, A316980, A316983, A318099, A319558, A319616-A319646, A300913.

Formula

Euler transform of A319625.