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A316767 Number of series-reduced locally stable rooted trees whose leaves form the integer partition with Heinz number n.

%I A316767 #7 Jul 12 2018 20:24:36
%S A316767 0,1,1,1,1,1,1,2,1,1,1,3,1,1,1,5,1,3,1,3,1,1,1,8,1,1,2,3,1,4,1,10,1,1,
%T A316767 1,12,1,1,1,8,1,4,1,3,3,1,1,24,1,3,1,3,1,8,1,8,1,1,1,17,1,1,3,24,1,4,
%U A316767 1,3,1,4,1,39,1,1,3,3,1,4,1,24,5,1,1,17
%N A316767 Number of series-reduced locally stable rooted trees whose leaves form the integer partition with Heinz number n.
%C A316767 A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally stable if no branch is a submultiset of any other branch of the same root.
%C A316767 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A316767 The a(24) = 8 trees:
%e A316767   (1(1(12)))
%e A316767   (1(2(11)))
%e A316767   (2(1(11)))
%e A316767   (1(112))
%e A316767   (2(111))
%e A316767   (11(12))
%e A316767   (12(11))
%e A316767   (1112)
%t A316767 sps[{}]:={{}};
%t A316767 sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A316767 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A316767 stableQ[u_]:=Apply[And,Outer[#1==#2||Complement[#2,#1]=!={}&,u,u,1],{0,1}];
%t A316767 gro[m_]:=gro[m]=If[Length[m]==1,List/@m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
%t A316767 Table[Length[Select[gro[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],And@@Cases[#,q:{__List}:>stableQ[q],{0,Infinity}]&]],{n,100}]
%Y A316767 Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316475, A316651, A316652, A316655.
%K A316767 nonn
%O A316767 1,8
%A A316767 _Gus Wiseman_, Jul 12 2018