A294079 - OEIS

%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