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A294079
Strict Moebius function of the multiorder of integer partitions indexed by Heinz numbers.
5
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
OFFSET
1,42
COMMENTS
By convention a(1) = 0.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
FORMULA
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.
MATHEMATICA
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
KEYWORD
sign
AUTHOR
Gus Wiseman, Feb 07 2018
STATUS
approved