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Number of non-isomorphic self-dual multiset partitions of weight n whose part sizes are relatively prime.
2

%I #4 Nov 16 2018 07:48:59

%S 1,1,1,3,6,16,27,71,135,309,621

%N Number of non-isomorphic self-dual multiset partitions of weight n whose part sizes 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).

%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(5) = 16 multiset partitions:

%e {{1}} {{1}{2}} {{1}{22}} {{1}{222}} {{11}{122}}

%e {{2}{12}} {{2}{122}} {{11}{222}}

%e {{1}{2}{3}} {{1}{1}{23}} {{12}{122}}

%e {{1}{2}{33}} {{1}{2222}}

%e {{1}{3}{23}} {{2}{1222}}

%e {{1}{2}{3}{4}} {{1}{22}{33}}

%e {{1}{23}{23}}

%e {{1}{2}{333}}

%e {{1}{3}{233}}

%e {{2}{12}{33}}

%e {{2}{13}{23}}

%e {{3}{3}{123}}

%e {{1}{2}{2}{34}}

%e {{1}{2}{3}{44}}

%e {{1}{2}{4}{34}}

%e {{1}{2}{3}{4}{5}}

%Y Cf. A000219, A007716, A120733, A138178, A316983, A319616.

%Y Cf. A320796, A320797, A320800, A320805, A320806, A320809, A320811, A320813, A321283, A321410, A321411, A321413.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Nov 16 2018