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A299202
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Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.
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32
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0, 1, 1, -1, 1, -1, 1, 0, -1, -1, 1, 2, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, 1, -1, -1, -1, 0, 1, 1, 3, 1, 0, -1, -1, -1, -1, 1, -1, -1, -1, 1, 2, 1, 1, 1, -1, 1, 0, -1, 1, -1, 1, 1, -1, -1, -1, -1, -1, 1, -3, 1, -1, 2, 0, -1, 2, 1, 1, -1, 3, 1, 2, 1, -1, 1, 1, -1, 2, 1, 1, -1, -1, 1, -5, -1, -1, -1, -1, 1, -4
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OFFSET
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1,12
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COMMENTS
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By convention, mu() = 0.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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LINKS
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FORMULA
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mu(y) = Sum_{g(t)=y} (-1)^d(t), where the sum is over all enriched p-trees (A289501, A299203) whose multiset of leaves is the integer partition y, and d(t) is the number of non-leaf nodes in t.
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EXAMPLE
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Heinz number of (2,1,1) is 12, so mu(2,1,1) = a(12) = 2.
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MATHEMATICA
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nn=120;
ptns=Table[If[n===1, {}, Join@@Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]], {n, nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&, ptns];
mu[y_]:=mu[y]=If[Length[y]===1, 1, -Sum[Times@@mu/@t, {t, Select[tris, And[Length[#]>1, Sort[Join@@#, Greater]===y]&]}]];
mu/@ptns
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CROSSREFS
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Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299201, A299203.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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