%I #4 Nov 16 2018 07:49:15
%S 1,0,0,0,0,1,0,4,6,16,25
%N Number of non-isomorphic self-dual multiset partitions of weight n with no singletons, with aperiodic parts whose sizes are relatively prime.
%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 relatively prime row sums (or column sums) and 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(9) = 16 multiset partitions:
%e {{12}{122}} {{112}{1222}} {{112}{12222}} {{1112}{11222}}
%e {{12}{12222}} {{122}{11222}} {{1112}{12222}}
%e {{12}{13}{233}} {{12}{123}{233}} {{12}{1222222}}
%e {{13}{23}{123}} {{13}{112}{233}} {{12}{123}{2333}}
%e {{13}{122}{233}} {{12}{13}{23333}}
%e {{23}{123}{123}} {{12}{223}{1233}}
%e {{13}{112}{2333}}
%e {{13}{223}{1233}}
%e {{13}{23}{12333}}
%e {{23}{122}{1233}}
%e {{23}{123}{1233}}
%e {{12}{12}{34}{234}}
%e {{12}{12}{34}{344}}
%e {{12}{13}{14}{234}}
%e {{12}{13}{24}{344}}
%e {{12}{14}{34}{234}}
%Y Cf. A000219, A007716, A120733, A138178, A302545, A316983, A319616.
%Y Cf. A320796, A320797, A320803, A320806, A320809, A320813, A321283, A321408-A321413.
%K nonn,more
%O 0,8
%A _Gus Wiseman_, Nov 16 2018
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