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

0,4

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.

Links

Table of n, a(n) for n=0..10.

Example

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.

Sequence in context: A305341 A316611 A174939 * A321315 A203225 A308386

Adjacent sequences: A320797 A320798 A320799 * A320801 A320802 A320803

Keyword

nonn,more

Author

Gus Wiseman, Nov 02 2018

Status

approved