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A368097
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Number of non-isomorphic multiset partitions of weight n contradicting a strict version of the axiom of choice.
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39
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0, 0, 1, 3, 12, 37, 133, 433, 1516, 5209, 18555
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OFFSET
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0,4
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COMMENTS
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A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.
The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
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LINKS
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EXAMPLE
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Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 multiset partitions:
{{1},{1}} {{1},{1,1}} {{1},{1,1,1}}
{{1},{1},{1}} {{1,1},{1,1}}
{{1},{2},{2}} {{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{1},{2,3}}
{{1},{2},{1,2}}
{{1},{2},{2,2}}
{{2},{2},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{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]}]]];
Table[Length[Union[brute/@Select[mpm[n], Select[Tuples[#], UnsameQ@@#&]=={}&]]], {n, 0, 6}]
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CROSSREFS
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The case of unlabeled graphs appears to be A140637, complement A134964.
These multiset partitions have ranks A355529.
Minimal multiset partitions of this type are ranked by A368187.
Factorizations of this type are counted by A368413, complement A368414.
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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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