A319643 Number of non-isomorphic weight-n strict multiset partitions whose dual is an antichain of (not necessarily distinct) multisets.

1, 1, 3, 6, 15, 29, 82, 179, 504, 1302, 3822

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.

From Gus Wiseman, Aug 15 2019: (Start)

Also the number of non-isomorphic T_0 weak antichains of weight n. The T_0 condition means that the dual is strict (no repeated edges). A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other. For example, non-isomorphic representatives of the a(0) = 1 through a(4) = 15 T_0 weak antichains are:

{} {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}

{{1},{1}} {{1,2,2}} {{1,2,2,2}}

{{1},{2}} {{1},{2,2}} {{1,1},{1,1}}

{{1},{1},{1}} {{1,1},{2,2}}

{{1},{2},{2}} {{1},{2,2,2}}

{{1},{2},{3}} {{1,2},{2,2}}

{{1},{2,3,3}}

{{1,3},{2,3}}

{{1},{1},{2,2}}

{{1},{2},{3,3}}

{{1},{1},{1},{1}}

{{1},{1},{2},{2}}

{{1},{2},{2},{2}}

{{1},{2},{3},{3}}

{{1},{2},{3},{4}}

(End)

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,3}}
   {{1},{1,1}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{2},{3}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{1},{2,2,2}}
   {{1},{2,3,4}}
   {{1,1},{2,2}}
   {{1,2},{3,3}}
   {{1,2},{3,4}}
   {{1},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{2},{3},{4}}
		

Crossrefs

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

Cf. A245567, A293993, A319721, A326704, A326950, A326973, A326978.