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

1,8

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).

Links

Table of n, a(n) for n=1..84.

Example

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

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}]

Crossrefs

Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316475, A316651, A316652, A316655.

Sequence in context: A326153 A199515 A316695 * A292505 A281119 A317176

Adjacent sequences: A316764 A316765 A316766 * A316768 A316769 A316770

Keyword

nonn

Author

Gus Wiseman, Jul 12 2018

Status

approved