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A320809
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Number of non-isomorphic multiset partitions of weight n in which each part and each part of the dual, as well as the multiset union of the parts, is an aperiodic multiset.
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5
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1, 1, 2, 5, 13, 40, 99, 344, 985, 3302, 10583
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OFFSET
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0,3
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COMMENTS
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Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which (1) the positive entries in each row and column are relatively prime and (2) the column sums are relatively prime.
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}}.
A multiset is aperiodic if its multiplicities are relatively prime.
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(4) = 13 multiset partitions:
{{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
{{1},{2}} {{1},{2,3}} {{1},{2,3,4}}
{{2},{1,2}} {{1,2},{3,4}}
{{1},{2},{2}} {{1,3},{2,3}}
{{1},{2},{3}} {{2},{1,2,2}}
{{3},{1,2,3}}
{{1},{1},{2,3}}
{{1},{2},{3,4}}
{{1},{3},{2,3}}
{{2},{2},{1,2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}
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CROSSREFS
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Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A316983, A320800-A320810, A321283.
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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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