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%I #4 Nov 18 2018 15:05:42
%S 1,0,0,0,1,1,3,4,12,20,42
%N Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and with aperiodic parts.
%C A multiset is aperiodic if its multiplicities are relatively prime.
%C Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with no row or column having a common divisor > 1 or summing to 1.
%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(5) = 1 through a(8) = 12 multiset partitions:
%e {{12}{12}} {{12}{122}} {{112}{122}} {{112}{1222}} {{1112}{1222}}
%e {{12}{1222}} {{12}{12222}} {{112}{12222}}
%e {{12}{13}{23}} {{12}{13}{233}} {{12}{122222}}
%e {{13}{23}{123}} {{122}{11222}}
%e {{12}{123}{233}}
%e {{12}{13}{2333}}
%e {{13}{112}{233}}
%e {{13}{122}{233}}
%e {{13}{23}{1233}}
%e {{23}{123}{123}}
%e {{12}{12}{34}{34}}
%e {{12}{13}{24}{34}}
%Y Cf. A000219, A007716, A120733, A138178, A302545, A316983, A319616.
%Y Cf. A320796, A320797, A320803, A320806, A320809, A320813, A321408, A321410, A321411.
%K nonn,more
%O 0,7
%A _Gus Wiseman_, Nov 16 2018
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