%I #10 Feb 23 2018 11:08:30
%S 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,
%T 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,
%U -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
%N Strict Moebius function of the multiorder of integer partitions indexed by Heinz numbers.
%C By convention a(1) = 0.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%F 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.
%t nn=120;
%t ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
%t tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
%t qmu[y_]:=qmu[y]=If[Length[y]===1,1,-Sum[Times@@qmu/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y,UnsameQ@@#]&]}]];
%t qmu/@ptns
%Y Cf. A000041, A000720, A056239, A063834, A196545, A273873, A289501, A294018, A294019, A296150, A299201, A299202, A299203.
%K sign
%O 1,42
%A _Gus Wiseman_, Feb 07 2018