A321390 Third Moebius transform of A007716. Number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods and whose dual is also an aperiodic multiset partition.

1, 1, 1, 7, 24, 88, 265, 907, 2929, 9918, 33931, 119366, 428314, 1574221, 5913415, 22699536, 88994103, 356058537, 1453049451, 6044132791, 25612496016, 110503624870, 485160989937, 2166488899639, 9835208617114, 45370059225048

Offset

0,4

Comments

The Moebius transform c of a sequence b is c(n) = Sum_{d|n} mu(d) * b(n/d).

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 and the nonzero entries are relatively prime, up to row and column permutations.

A multiset is aperiodic if its multiplicities are relatively prime. The period of a multiset is the GCD of its multiplicities.

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..25.

Example

Non-isomorphic representatives of the a(1) = 1 through a(4) = 24 multiset partitions:
  {{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},{2,2,2}}
                    {{1},{2},{2}}  {{1,2},{2,2}}
                    {{1},{2},{3}}  {{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: A196349 A196352 A050191 * A283457 A129797 A188120

Adjacent sequences: A321387 A321388 A321389 * A321391 A321392 A321393

Keyword

nonn

Author

Gus Wiseman, Nov 08 2018

Status

approved