OFFSET
1,2
COMMENTS
A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (unequal) child of the same vertex. Lone-child-avoiding means there are no unary branchings. In an identity tree, all branches of any given vertex are distinct.
EXAMPLE
The a(1) = 1 through a(5) = 17 trees:
(1) (2) (3) (4) (5)
(11) (12) (13) (14)
(111) (22) (23)
((1)(2)) (112) (113)
(1111) (122)
((1)(3)) (1112)
((2)(11)) (11111)
((1)((1)(2))) ((1)(4))
((2)(3))
((1)(22))
((3)(11))
((2)(111))
((1)((1)(3)))
((2)((1)(2)))
((11)((1)(2)))
((1)((2)(11)))
((1)((1)((1)(2))))
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}];
mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]], UnsameQ@@#&&disjointQ[#]&], {p, Select[mps[m], Length[#]>1&]}], m];
Table[Sum[Length[mpti[m]], {m, Sort/@IntegerPartitions[n]}], {n, 8}]
CROSSREFS
The non-identity version is A331678.
The case where the leaves are all singletons is A316694.
Identity trees are A004111.
Locally disjoint identity trees are A316471.
Locally disjoint enriched identity p-trees are A331684.
Lone-child-avoiding locally disjoint rooted semi-identity trees are A212804.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jan 31 2020
STATUS
approved