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A299202 Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.

%I A299202 #15 Feb 23 2018 11:10:35
%S A299202 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 A299202 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 A299202 -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 A299202 Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.
%C A299202 By convention, mu() = 0.
%C A299202 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A299202 Gus Wiseman, <a href="https://docs.google.com/document/d/1m0s6DGTBkDW9gvMuFmJHvy6oLGRAbQ7okAZcOPZawp0/pub">Comcategories and Multiorders</a>
%F A299202 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 A299202 Heinz number of (2,1,1) is 12, so mu(2,1,1) = a(12) = 2.
%t A299202 nn=120;
%t A299202 ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
%t A299202 tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
%t A299202 mu[y_]:=mu[y]=If[Length[y]===1,1,-Sum[Times@@mu/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y]&]}]];
%t A299202 mu/@ptns
%Y A299202 Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299201, A299203.
%K A299202 sign
%O A299202 1,12
%A A299202 _Gus Wiseman_, Feb 05 2018