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%I #5 Nov 07 2018 21:45:25
%S 1,1,2,6,16,55,139,516,1500,5269,17017
%N Number of non-isomorphic multiset partitions of weight n in which each part, as well as the multiset union of the parts, is an aperiodic multiset.
%C Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which (1) the positive entries in each row are relatively prime and (2) the column sums are relatively prime.
%C A multiset is aperiodic if its multiplicities are relatively prime.
%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) = 16 multiset partitions:
%e {{1}} {{1,2}} {{1,2,2}} {{1,2,2,2}}
%e {{1},{2}} {{1,2,3}} {{1,2,3,3}}
%e {{1},{2,3}} {{1,2,3,4}}
%e {{2},{1,2}} {{1},{2,3,3}}
%e {{1},{2},{2}} {{1},{2,3,4}}
%e {{1},{2},{3}} {{1,2},{3,4}}
%e {{1,3},{2,3}}
%e {{2},{1,2,2}}
%e {{3},{1,2,3}}
%e {{1},{1},{2,3}}
%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, A303546, A303707, A303708, A303710, A320800-A320810, A321283.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Nov 07 2018
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