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A321408
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Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic.
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3
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1, 1, 1, 2, 5, 9, 18, 35, 75, 153, 318
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OFFSET
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0,4
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COMMENTS
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A multiset is aperiodic if its multiplicities are relatively prime.
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, in which no row or column has a common divisor > 1.
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}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
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LINKS
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EXAMPLE
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Non-isomorphic representatives of the a(1) = 1 through a(6) = 18 multiset partitions:
{1} {1}{2} {2}{12} {12}{12} {12}{122} {112}{122}
{1}{2}{3} {2}{122} {2}{1222} {12}{1222}
{1}{1}{23} {1}{23}{23} {2}{12222}
{1}{3}{23} {1}{3}{233} {12}{13}{23}
{1}{2}{3}{4} {2}{13}{23} {1}{23}{233}
{3}{3}{123} {1}{3}{2333}
{1}{2}{2}{34} {2}{13}{233}
{1}{2}{4}{34} {3}{23}{123}
{1}{2}{3}{4}{5} {3}{3}{1233}
{1}{1}{1}{234}
{1}{2}{34}{34}
{1}{2}{4}{344}
{1}{3}{24}{34}
{1}{4}{4}{234}
{2}{4}{12}{34}
{1}{2}{3}{3}{45}
{1}{2}{3}{5}{45}
{1}{2}{3}{4}{5}{6}
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CROSSREFS
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Cf. A320796, A320797, A320803, A320804, A320805, A320806, A320807, A320809, A320813, A321410, A321411, A321412.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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