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

1,12

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).

Links

Table of n, a(n) for n=1..86.

Example

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

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}]

Crossrefs

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

Sequence in context: A348653 A278287 A227752 * A306269 A035212 A318133

Adjacent sequences: A316768 A316769 A316770 * A316772 A316773 A316774

Keyword

nonn

Author

Gus Wiseman, Jul 12 2018

Status

approved