cp's OEIS Frontend

A319286 Number of series-reduced locally disjoint rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

%I A319286 #4 Sep 17 2018 08:34:40
%S A319286 1,2,9,67,573,6933,97147,1666999
%N A319286 Number of series-reduced locally disjoint rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
%C A319286 A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other branch of the same root.
%e A319286 The a(3) = 9 trees:
%e A319286   (1(11))
%e A319286    (111)
%e A319286   (1(12))
%e A319286   (2(11))
%e A319286    (112)
%e A319286   (1(23))
%e A319286   (2(13))
%e A319286   (3(12))
%e A319286    (123)
%e A319286 Examples of rooted trees that are not locally disjoint are ((11)(12)) and ((12)(13)).
%t A319286 disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t A319286 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A319286 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A319286 gro[m_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],disjointQ]];
%t A319286 Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]
%Y A319286 Cf. A000081, A007562, A316473, A316475, A316494, A316495, A316496, A316497.
%K A319286 nonn,more
%O A319286 1,2
%A A319286 _Gus Wiseman_, Sep 16 2018