cp's OEIS Frontend

A331686 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

%I A331686 #5 Feb 01 2020 14:39:45
%S A331686 1,2,4,8,17,41,103,280,793,2330,6979,21291
%N A331686 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 A331686 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 A331686 The a(1) = 1 through a(5) = 17 trees:
%e A331686   (1)  (2)   (3)       (4)            (5)
%e A331686        (11)  (12)      (13)           (14)
%e A331686              (111)     (22)           (23)
%e A331686              ((1)(2))  (112)          (113)
%e A331686                        (1111)         (122)
%e A331686                        ((1)(3))       (1112)
%e A331686                        ((2)(11))      (11111)
%e A331686                        ((1)((1)(2)))  ((1)(4))
%e A331686                                       ((2)(3))
%e A331686                                       ((1)(22))
%e A331686                                       ((3)(11))
%e A331686                                       ((2)(111))
%e A331686                                       ((1)((1)(3)))
%e A331686                                       ((2)((1)(2)))
%e A331686                                       ((11)((1)(2)))
%e A331686                                       ((1)((2)(11)))
%e A331686                                       ((1)((1)((1)(2))))
%t A331686 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A331686 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A331686 disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t A331686 mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]],UnsameQ@@#&&disjointQ[#]&],{p,Select[mps[m],Length[#]>1&]}],m];
%t A331686 Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]
%Y A331686 The non-identity version is A331678.
%Y A331686 The case where the leaves are all singletons is A316694.
%Y A331686 Identity trees are A004111.
%Y A331686 Locally disjoint identity trees are A316471.
%Y A331686 Locally disjoint enriched identity p-trees are A331684.
%Y A331686 Lone-child-avoiding locally disjoint rooted semi-identity trees are A212804.
%Y A331686 Cf. A000669, A001678, A005804, A141268, A300660, A316697, A319312, A331679, A331683, A331783, A331874, A331875.
%K A331686 nonn,more
%O A331686 1,2
%A A331686 _Gus Wiseman_, Jan 31 2020