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Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic multisets whose sizes are relatively prime.
4

%I #4 Nov 16 2018 07:49:07

%S 1,1,1,2,4,9,15,35,69,149,301

%N Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic multisets 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.

%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(6) = 15 multiset partitions:

%e {1} {1}{2} {2}{12} {2}{122} {12}{122} {2}{12222}

%e {1}{2}{3} {1}{1}{23} {2}{1222} {1}{23}{233}

%e {1}{3}{23} {1}{23}{23} {1}{3}{2333}

%e {1}{2}{3}{4} {1}{3}{233} {2}{13}{233}

%e {2}{13}{23} {3}{23}{123}

%e {3}{3}{123} {3}{3}{1233}

%e {1}{2}{2}{34} {1}{1}{1}{234}

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

%e {1}{2}{3}{4}{5} {1}{2}{4}{344}

%e {1}{3}{24}{34}

%e {1}{4}{4}{234}

%e {2}{4}{12}{34}

%e {1}{2}{3}{3}{45}

%e {1}{2}{3}{5}{45}

%e {1}{2}{3}{4}{5}{6}

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

%Y Cf. A320796, A320803, A320806, A320809, A320813, A321283, A321408, A321409, A321411.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Nov 16 2018