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%I #5 Sep 17 2018 08:34:47
%S 1,2,12,107,1299,20764,412957,9817743
%N Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.
%e The a(3) = 12 series-reduced locally disjoint rooted trees:
%e (1(11))
%e (111)
%e (1(22))
%e (2(12))
%e (122)
%e (1(12))
%e (2(11))
%e (112)
%e (1(23))
%e (2(13))
%e (3(12))
%e (123)
%e The trees counted by A316651(4) but not by a(4):
%e ((11)(12))
%e ((12)(13))
%e ((12)(22))
%e ((12)(23))
%e ((13)(23))
%t disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%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 gro[m_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],disjointQ]];
%t allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]
%Y Cf. A000081, A007562, A301700, A316473, A316475, A316495, A316651, A316694, A316695, A316696, A316697, A319286.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Sep 16 2018
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