A320802 Number of non-isomorphic aperiodic multiset partitions of weight n whose dual is also an aperiodic multiset partition.

1, 1, 2, 8, 26, 89, 274, 908, 2955, 9926, 34021, 119367, 428612, 1574222, 5914324, 22699632, 88997058, 356058538, 1453059643, 6044132792, 25612530061, 110503625785, 485161109305, 2166488899640, 9835209048655, 45370059225137, 212582814591083, 1011306624492831

Offset

0,3

Comments

Also the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns where the multiset of rows and the multiset of columns are both aperiodic, up to row and column permutations.

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.

Also the number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods, where the period of a multiset is the GCD of its multiplicities.

Links

Jinyuan Wang, Table of n, a(n) for n = 0..50

Formula

Second Moebius transform of A007716, or Moebius transform of A303546, where the Meobius transform of a sequence b is a(n) = Sum_{d|n} mu(d) * b(n/d).

Example

Non-isomorphic representatives of the a(1) = 1 through a(4) = 26 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1,2,2}}      {{1,2,2,2}}
                    {{1},{1,1}}    {{1,2,3,3}}
                    {{1},{2,2}}    {{1},{1,1,1}}
                    {{1},{2,3}}    {{1},{1,2,2}}
                    {{2},{1,2}}    {{1,1},{2,2}}
                    {{1},{2},{2}}  {{1},{2,2,2}}
                    {{1},{2},{3}}  {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{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, A320800-A320810.

Sequence in context: A343485 A298189 A053956 * A052543 A026638 A307401

Adjacent sequences: A320799 A320800 A320801 * A320803 A320804 A320805

Keyword

nonn

Author

Gus Wiseman, Nov 06 2018

Extensions

a(26)-a(27) from Jinyuan Wang, Jun 27 2020

Status

approved