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.

1, 2, 4, 8, 17, 41, 103, 280, 793, 2330, 6979, 21291

Offset: 1

Author

Gus Wiseman, Jan 31 2020

Keywords

• nonn
• more

Comments

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.

Examples

			The a(1) = 1 through a(5) = 17 trees:
  (1)  (2)   (3)       (4)            (5)
       (11)  (12)      (13)           (14)
             (111)     (22)           (23)
             ((1)(2))  (112)          (113)
                       (1111)         (122)
                       ((1)(3))       (1112)
                       ((2)(11))      (11111)
                       ((1)((1)(2)))  ((1)(4))
                                      ((2)(3))
                                      ((1)(22))
                                      ((3)(11))
                                      ((2)(111))
                                      ((1)((1)(3)))
                                      ((2)((1)(2)))
                                      ((11)((1)(2)))
                                      ((1)((2)(11)))
                                      ((1)((1)((1)(2))))
		

Crossrefs

The non-identity version is A331678.

The case where the leaves are all singletons is A316694.

Identity trees are A004111.

Locally disjoint identity trees are A316471.

Locally disjoint enriched identity p-trees are A331684.

Lone-child-avoiding locally disjoint rooted semi-identity trees are A212804.

Cf. A000669, A001678, A005804, A141268, A300660, A316697, A319312, A331679, A331683, A331783, A331874, A331875.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]],UnsameQ@@#&&disjointQ[#]&],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]