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A316695 Number of series-reduced locally disjoint rooted trees whose leaves form the integer partition with Heinz number n. 1
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, 23, 1, 3, 1, 3, 1, 8, 1, 8, 1, 1, 1, 16, 1, 1, 3, 24, 1, 4, 1, 3, 1, 4, 1, 37, 1, 1, 3, 3, 1, 4, 1, 23, 5, 1, 1, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
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.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
LINKS
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]]]];
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, 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}:>disjointQ[q], {0, Infinity}]&]], {n, 100}]
CROSSREFS
Sequence in context: A326840 A326153 A199515 * A316767 A292505 A281119
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 10 2018
STATUS
approved

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Last modified March 26 23:33 EDT 2024. Contains 371192 sequences. (Running on oeis4.)