cp's OEIS Frontend

A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.

%I A331680 #15 Feb 01 2020 07:08:24
%S A331680 1,0,1,1,2,3,6,9,16,26,45,72,124,201,341,561,947,1571,2651,4434,7496,
%T A331680 12631,21423,36332,61910,105641,180924,310548,534713,923047
%N A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.
%C A331680 First differs from A320268 at a(11) = 45, A320268(11) = 44.
%C A331680 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.
%H A331680 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 A331680 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 A331680 The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):
%e A331680   o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
%e A331680                      (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
%e A331680                               (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
%e A331680                                         (ooo(oo))   (ooo(ooo))   (ooo(oooo))
%e A331680                                         ((oo)(oo))  (oooo(oo))   (oooo(ooo))
%e A331680                                         (o(o(oo)))  (o(o(ooo)))  (ooooo(oo))
%e A331680                                                     (o(oo)(oo))  ((ooo)(ooo))
%e A331680                                                     (o(oo(oo)))  (o(o(oooo)))
%e A331680                                                     (oo(o(oo)))  (o(oo(ooo)))
%e A331680                                                                  (o(ooo(oo)))
%e A331680                                                                  (oo(o(ooo)))
%e A331680                                                                  (oo(oo)(oo))
%e A331680                                                                  (oo(oo(oo)))
%e A331680                                                                  (ooo(o(oo)))
%e A331680                                                                  (o((oo)(oo)))
%e A331680                                                                  (o(o(o(oo))))
%t A331680 disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t A331680 strut[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@Rest[IntegerPartitions[n-1]],disjointQ]];
%t A331680 Table[Length[strut[n]],{n,10}]
%Y A331680 The enriched version is A316696.
%Y A331680 The Matula-Goebel numbers of these trees are A331871.
%Y A331680 The non-locally disjoint version is A001678.
%Y A331680 These trees counted by number of leaves are A316697.
%Y A331680 The semi-lone-child-avoiding version is A331872.
%Y A331680 Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.
%K A331680 nonn,more
%O A331680 1,5
%A A331680 _Gus Wiseman_, Jan 25 2020