A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.

1, 2, 4, 11, 27, 80, 218, 654, 1923, 5924, 18310, 58176, 186341, 606814, 1993420, 6618160, 22134640

Offset: 1

Author

Gus Wiseman, Jul 10 2018

Keywords

• nonn
• more

Comments

A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.

Examples

			The a(4) = 11 rooted trees:
  4,
  (13),
  (22),
  (1(12)), (2(11)), (112),
  (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111).
		

Links

• Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Crossrefs

Matula-Goebel numbers of locally disjoint rooted trees are A316495.

The case where all leaves are 1's is A316697.

Lone-child-avoiding locally disjoint rooted trees are A331680.

Cf. A000669, A001678, A141268, A316473, A316652, A331678, A331686, A331687, A331871, A331872, A331874.

Programs

Mathematica

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],disjointQ],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

Extensions

a(16)-a(17) from Robert Price, Sep 16 2018

Terminology corrected by Gus Wiseman, Feb 06 2020