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%I #5 Nov 05 2018 21:01:23
%S 1,1,1,5,14,78,157,881,2267,9257,28397
%N 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.
%C The latter condition is equivalent to the parts having relatively prime sizes.
%C A multiset is aperiodic if its multiplicities are relatively prime.
%C 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}}.
%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 14 multiset partitions:
%e {{1}} {{1},{2}} {{1},{2,2}} {{1},{2,2,2}}
%e {{1},{2,3}} {{1},{2,3,3}}
%e {{2},{1,2}} {{1},{2,3,4}}
%e {{1},{2},{2}} {{2},{1,2,2}}
%e {{1},{2},{3}} {{3},{1,2,3}}
%e {{1},{1},{2,3}}
%e {{1},{2},{2,2}}
%e {{1},{2},{3,3}}
%e {{1},{2},{3,4}}
%e {{1},{3},{2,3}}
%e {{2},{2},{1,2}}
%e {{1},{2},{2},{2}}
%e {{1},{2},{3},{3}}
%e {{1},{2},{3},{4}}
%Y Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A316983, A320801-A320810.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Nov 02 2018
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