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%I #13 Jan 30 2020 08:42:03
%S 1,3,6,18,44,149,450,1573,5352,19283,69483,257206
%N Number of lone-child-avoiding locally disjoint rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.
%C Lone-child-avoiding means there are no unary branchings. Locally disjoint means no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex.
%H 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 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 The a(1) = 1 through a(4) = 18 trees:
%e (1) (2) (3) (4)
%e (11) (12) (13)
%e ((1)(1)) (111) (22)
%e ((1)(2)) (112)
%e ((1)(1)(1)) (1111)
%e ((1)((1)(1))) ((1)(3))
%e ((2)(2))
%e ((2)(11))
%e ((11)(11))
%e ((1)(1)(2))
%e ((1)((1)(2)))
%e ((2)((1)(1)))
%e ((1)(1)(1)(1))
%e ((11)((1)(1)))
%e ((1)((1)(1)(1)))
%e ((1)(1)((1)(1)))
%e (((1)(1))((1)(1)))
%e ((1)((1)((1)(1))))
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]],disjointQ],{p,Select[mps[m],Length[#]>1&]}],m];
%t Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]
%Y The case where all leaves are singletons is A316696.
%Y The case where all leaves are (1) is A316697.
%Y The non-locally disjoint version is A319312.
%Y The case with all atoms equal to 1 is A331679.
%Y The identity tree case is A331686.
%Y Cf. A000081, A000669, A001678, A005804, A060356, A141268, A196545, A300660, A316471, A316694, A316495, A330465, A331680, A331687.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Jan 25 2020
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