OFFSET
1,2
COMMENTS
A tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. It is lone-child-avoiding if there are no unary branchings.
LINKS
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
EXAMPLE
The a(1) = 1 through a(5) = 16 trees:
1 2 3 4 5
(11) (111) (22) (11111)
(1(11)) (1111) ((11)3)
(2(11)) (1(22))
(1(111)) (2(111))
(11(11)) (1(1111))
((11)(11)) (11(111))
(1(1(11))) (111(11))
(1(2(11)))
(2(1(11)))
(1(1(111)))
(1(11)(11))
(1(11(11)))
(11(1(11)))
(1((11)(11)))
(1(1(1(11))))
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
usot[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[usot/@ptn]], disjointQ[DeleteCases[#, _?AtomQ]]&&SameQ@@Select[#, AtomQ]&], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}], n];
Table[Length[usot[n]], {n, 12}]
CROSSREFS
The non-locally disjoint version is A141268.
Locally disjoint trees counted by vertices are A316473.
The case where all leaves are 1's is A316697.
Number of trees counted by A331678 with all atoms equal to 1.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Unlabeled lone-child-avoiding locally disjoint rooted trees are A331680.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jan 25 2020
STATUS
approved