A294079 Strict Moebius function of the multiorder of integer partitions indexed by Heinz numbers.

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

Author

Gus Wiseman, Feb 07 2018

Keywords

• sign

Comments

By convention a(1) = 0.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Crossrefs

Cf. A000041, A000720, A056239, A063834, A196545, A273873, A289501, A294018, A294019, A296150, A299201, A299202, A299203.

Programs

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

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.