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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
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).
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 10 2018
STATUS
approved