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A294079
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Strict Moebius function of the multiorder of integer partitions indexed by Heinz numbers.
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5
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0, 1, 1, 0, 1, -1, 1, 0, 0, -1, 1, 1, 1, -1, -1, 0, 1, 1, 1, 1, -1, -1, 1, -1, 0, -1, 0, 1, 1, 1, 1, 0, -1, -1, -1, -1, 1, -1, -1, -1, 1, 2, 1, 1, 1, -1, 1, 1, 0, 1, -1, 1, 1, -1, -1, -1, -1, -1, 1, -3, 1, -1, 1, 0, -1, 2, 1, 1, -1, 1, 1, 2, 1, -1, 1, 1, -1, 2, 1, 1, 0, -1, 1, -2, -1, -1, -1, -1, 1, -3, -1
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OFFSET
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1,42
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COMMENTS
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By convention a(1) = 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 strict trees (A273873) 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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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];
qmu[y_]:=qmu[y]=If[Length[y]===1, 1, -Sum[Times@@qmu/@t, {t, Select[tris, And[Length[#]>1, Sort[Join@@#, Greater]===y, UnsameQ@@#]&]}]];
qmu/@ptns
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CROSSREFS
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Cf. A000041, A000720, A056239, A063834, A196545, A273873, A289501, A294018, A294019, A296150, A299201, A299202, 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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