A320800 Number of non-isomorphic multiset partitions of weight n in which both the multiset union of the parts and the multiset union of the dual parts are aperiodic.

1, 1, 1, 5, 14, 78, 157, 881, 2267, 9257, 28397

Offset: 0

Author

Gus Wiseman, Nov 02 2018

Keywords

• nonn
• more

Comments

The latter condition is equivalent to the parts having relatively prime sizes.

A multiset is aperiodic if its multiplicities are relatively prime.

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts 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(4) = 14 multiset partitions:
  {{1}}  {{1},{2}}  {{1},{2,2}}    {{1},{2,2,2}}
                    {{1},{2,3}}    {{1},{2,3,3}}
                    {{2},{1,2}}    {{1},{2,3,4}}
                    {{1},{2},{2}}  {{2},{1,2,2}}
                    {{1},{2},{3}}  {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}
		

Crossrefs

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A316983, A320801-A320810.