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A316767
Number of series-reduced locally stable rooted trees whose leaves form the integer partition with Heinz number n.
0
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).
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 12 2018
STATUS
approved