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A319291
Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.
0
1, 2, 12, 107, 1299, 20764, 412957, 9817743
OFFSET
1,2
EXAMPLE
The a(3) = 12 series-reduced locally disjoint rooted trees:
(1(11))
(111)
(1(22))
(2(12))
(122)
(1(12))
(2(11))
(112)
(1(23))
(2(13))
(3(12))
(123)
The trees counted by A316651(4) but not by a(4):
((11)(12))
((12)(13))
((12)(22))
((12)(23))
((13)(23))
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
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]]]];
gro[m_]:=gro[m]=If[Length[m]==1, {m}, Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])], disjointQ]];
allnorm[n_Integer]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]], {m, allnorm[n]}], {n, 5}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 16 2018
STATUS
approved