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A320813
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Number of non-isomorphic multiset partitions of an aperiodic multiset of weight n such that there are no singletons and all parts are themselves aperiodic multisets.
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13
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1, 0, 1, 2, 5, 13, 33, 104, 293, 938, 2892
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OFFSET
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0,4
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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 row sums are all > 1, (2) the positive entries in each row are relatively prime, and (3) the column-sums are relatively prime.
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(2) = 1 through a(5) = 13 multiset partitions:
{{1,2}} {{1,2,2}} {{1,2,2,2}} {{1,1,2,2,2}}
{{1,2,3}} {{1,2,3,3}} {{1,2,2,2,2}}
{{1,2,3,4}} {{1,2,2,3,3}}
{{1,2},{3,4}} {{1,2,3,3,3}}
{{1,3},{2,3}} {{1,2,3,4,4}}
{{1,2,3,4,5}}
{{1,2},{1,2,2}}
{{1,2},{2,3,3}}
{{1,2},{3,4,4}}
{{1,2},{3,4,5}}
{{1,3},{2,3,3}}
{{1,4},{2,3,4}}
{{2,3},{1,2,3}}
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s, Flatten[MapIndexed[Table[#2, {#1}]&, #]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i, p[[i]]}, {i, Length[p]}])], {p, Permutations[Union@@m]}]]];
aperQ[m_]:=Length[m]==0||GCD@@Length/@Split[Sort[m]]==1;
Table[Length[Union[brute /@ Select[mpm[n], And[Min@@Length/@#>1, aperQ[Join@@#]&&And@@aperQ /@ #]&]]], {n, 0, 7}] (* Gus Wiseman, Jan 19 2024 *)
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CROSSREFS
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This is the case of A320804 where the underlying multiset is aperiodic.
Cf. A000740, A000837, A007716, A007916, A100953, A301700, A302545, A303386, A303546, A303707, A303708, A320797-A320813, A321283, A321390.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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