cp's OEIS Frontend

A331679 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.

%I A331679 #15 Jan 30 2020 08:42:13
%S A331679 1,2,3,8,16,48,116,341,928,2753,7996,24254,73325,226471,702122
%N A331679 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.
%C A331679 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.
%H A331679 David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], (30-June-2014).
%H A331679 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>
%e A331679 The a(1) = 1 through a(5) = 16 trees:
%e A331679   1  2     3        4           5
%e A331679      (11)  (111)    (22)        (11111)
%e A331679            (1(11))  (1111)      ((11)3)
%e A331679                     (2(11))     (1(22))
%e A331679                     (1(111))    (2(111))
%e A331679                     (11(11))    (1(1111))
%e A331679                     ((11)(11))  (11(111))
%e A331679                     (1(1(11)))  (111(11))
%e A331679                                 (1(2(11)))
%e A331679                                 (2(1(11)))
%e A331679                                 (1(1(111)))
%e A331679                                 (1(11)(11))
%e A331679                                 (1(11(11)))
%e A331679                                 (11(1(11)))
%e A331679                                 (1((11)(11)))
%e A331679                                 (1(1(1(11))))
%t A331679 disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t A331679 usot[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[usot/@ptn]],disjointQ[DeleteCases[#,_?AtomQ]]&&SameQ@@Select[#,AtomQ]&],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
%t A331679 Table[Length[usot[n]],{n,12}]
%Y A331679 The non-locally disjoint version is A141268.
%Y A331679 Locally disjoint trees counted by vertices are A316473.
%Y A331679 The case where all leaves are 1's is A316697.
%Y A331679 Number of trees counted by A331678 with all atoms equal to 1.
%Y A331679 Matula-Goebel numbers of locally disjoint rooted trees are A316495.
%Y A331679 Unlabeled lone-child-avoiding locally disjoint rooted trees are A331680.
%Y A331679 Cf. A000081, A000669, A001678, A005804, A060356, A300660, A316471, A316694, A316696, A319312, A330465, A331681.
%K A331679 nonn,more
%O A331679 1,2
%A A331679 _Gus Wiseman_, Jan 25 2020