%I #5 Feb 01 2020 14:39:45
%S 1,2,4,8,17,41,103,280,793,2330,6979,21291
%N Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.
%C 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.
%e The a(1) = 1 through a(5) = 17 trees:
%e (1) (2) (3) (4) (5)
%e (11) (12) (13) (14)
%e (111) (22) (23)
%e ((1)(2)) (112) (113)
%e (1111) (122)
%e ((1)(3)) (1112)
%e ((2)(11)) (11111)
%e ((1)((1)(2))) ((1)(4))
%e ((2)(3))
%e ((1)(22))
%e ((3)(11))
%e ((2)(111))
%e ((1)((1)(3)))
%e ((2)((1)(2)))
%e ((11)((1)(2)))
%e ((1)((2)(11)))
%e ((1)((1)((1)(2))))
%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]],UnsameQ@@#&&disjointQ[#]&],{p,Select[mps[m],Length[#]>1&]}],m];
%t Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]
%Y The non-identity version is A331678.
%Y The case where the leaves are all singletons is A316694.
%Y Identity trees are A004111.
%Y Locally disjoint identity trees are A316471.
%Y Locally disjoint enriched identity p-trees are A331684.
%Y Lone-child-avoiding locally disjoint rooted semi-identity trees are A212804.
%Y Cf. A000669, A001678, A005804, A141268, A300660, A316697, A319312, A331679, A331683, A331783, A331874, A331875.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Jan 31 2020