A320807 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic and all parts of the dual are also aperiodic.

1, 1, 3, 6, 17, 41, 122, 345, 1077, 3385, 11214

Offset

0,3

Comments

Also the number of nonnegative integer matrices up to row and column permutations with sum of entries equal to n and no zero rows or columns, in which each row and each column has relatively prime nonzero entries.

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}}.

A multiset is aperiodic if its multiplicities are relatively prime.

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) = 17 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}
         {{1},{2}}  {{2},{1,2}}    {{1},{2,3,4}}
                    {{1},{1},{1}}  {{1,2},{3,4}}
                    {{1},{2},{2}}  {{1,3},{2,3}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Crossrefs

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A316983, A320800-A320810.

Sequence in context: A007718 A297972 A275057 * A089264 A121399 A212421

Adjacent sequences: A320804 A320805 A320806 * A320808 A320809 A320810

Keyword

nonn,more

Author

Gus Wiseman, Nov 07 2018

Status

approved