A316771 Number of series-reduced locally nonintersecting rooted trees whose leaves form the integer partition with Heinz number n.

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 4, 0, 1, 0, 2, 1, 4, 1, 0, 1, 1, 1, 6, 1, 1, 1, 4, 1, 4, 1, 2, 2, 1, 1, 8, 0, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 12, 1, 1, 2, 0, 1, 4, 1, 2, 1, 4, 1, 17, 1, 1, 2, 2, 1, 4, 1, 8, 0, 1, 1, 12, 1, 1

Offset: 1

Author

Gus Wiseman, Jul 12 2018

Keywords

• nonn

Comments

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given root is empty.

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

Examples

			The a(36) = 6 trees:
  (1(2(12)))
  (2(1(12)))
  (1(122))
  (2(112))
  (12(12))
  (1122)
		

Crossrefs

Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316503, A316651, A316652, A316655.

Programs

Mathematica

sps[{}]:={{}};
sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=gro[m]=If[Length[m]==1,List/@m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[Select[gro[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],And@@Cases[#,q:{__List}:>Intersection@@q=={},{0,Infinity}]&]],{n,100}]