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

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, 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, 1, 3, 1, 4, 1, 39, 1, 1, 3, 3, 1, 4, 1, 24, 5, 1, 1, 17

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 stable if no branch is a submultiset of any other branch of the same root.

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

Examples

			The a(24) = 8 trees:
  (1(1(12)))
  (1(2(11)))
  (2(1(11)))
  (1(112))
  (2(111))
  (11(12))
  (12(11))
  (1112)
		

Crossrefs

Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316475, 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]]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||Complement[#2,#1]=!={}&,u,u,1],{0,1}];
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}:>stableQ[q],{0,Infinity}]&]],{n,100}]