cp's OEIS Frontend

A316695 Number of series-reduced locally disjoint rooted trees whose leaves form the integer partition with Heinz number n.

%I A316695 #6 Jul 10 2018 21:15:39
%S A316695 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 A316695 1,12,1,1,1,8,1,4,1,3,3,1,1,23,1,3,1,3,1,8,1,8,1,1,1,16,1,1,3,24,1,4,
%U A316695 1,3,1,4,1,37,1,1,3,3,1,4,1,23,5,1,1,16
%N A316695 Number of series-reduced locally disjoint rooted trees whose leaves form the integer partition with Heinz number n.
%C A316695 A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.
%C A316695 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A316695 The a(24) = 8 trees:
%e A316695   (1(1(12)))
%e A316695   (1(2(11)))
%e A316695   (2(1(11)))
%e A316695   (1(112))
%e A316695   (2(111))
%e A316695   (11(12))
%e A316695   (12(11))
%e A316695   (1112)
%t A316695 sps[{}]:={{}};
%t A316695 sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A316695 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A316695 disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t A316695 gro[m_]:=gro[m]=If[Length[m]==1,List/@m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
%t A316695 Table[Length[Select[gro[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],And@@Cases[#,q:{__List}:>disjointQ[q],{0,Infinity}]&]],{n,100}]
%Y A316695 Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316471, A316651, A316652, A316655.
%K A316695 nonn
%O A316695 1,8
%A A316695 _Gus Wiseman_, Jul 10 2018