A299202 - OEIS

%I #15 Feb 23 2018 11:10:35

%S 0,1,1,-1,1,-1,1,0,-1,-1,1,2,1,-1,-1,-1,1,1,1,1,-1,-1,1,-1,-1,-1,0,1,

%T 1,3,1,0,-1,-1,-1,-1,1,-1,-1,-1,1,2,1,1,1,-1,1,0,-1,1,-1,1,1,-1,-1,-1,

%U -1,-1,1,-3,1,-1,2,0,-1,2,1,1,-1,3,1,2,1,-1,1,1,-1,2,1,1,-1,-1,1,-5,-1,-1,-1,-1,1,-4

%N Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.

%C By convention, mu() = 0.

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

%H Gus Wiseman, <a href="https://docs.google.com/document/d/1m0s6DGTBkDW9gvMuFmJHvy6oLGRAbQ7okAZcOPZawp0/pub">Comcategories and Multiorders</a>

%F mu(y) = Sum_{g(t)=y} (-1)^d(t), where the sum is over all enriched p-trees (A289501, A299203) whose multiset of leaves is the integer partition y, and d(t) is the number of non-leaf nodes in t.

%e Heinz number of (2,1,1) is 12, so mu(2,1,1) = a(12) = 2.

%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 mu[y_]:=mu[y]=If[Length[y]===1,1,-Sum[Times@@mu/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y]&]}]];

%t mu/@ptns

%Y Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299201, A299203.

%K sign

%O 1,12

%A _Gus Wiseman_, Feb 05 2018