OFFSET
1,5
COMMENTS
An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.
LINKS
EXAMPLE
The a(11) = 12 locally connected rooted trees:
((((((((((o))))))))))
((((((((o)(o))))))))
(((((((o))((o)))))))
((((((o)))(((o))))))
(((((o))))((((o)))))
((((((o)(o)(o))))))
(((((o))((o)(o)))))
((((o))((o))((o))))
((((o)(o)(o)(o))))
(((o))((o)(o)(o)))
(((o)(o))((o)(o)))
((o)(o)(o)(o)(o))
MATHEMATICA
multijoin[mss__]:=Join@@Table[Table[x, {Max[Count[#, x]&/@{mss}]}], {x, Union[mss]}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], multijoin@@s[[c[[1]]]]]]]]];
rurt[n_]:=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]], Or[Length[#]==1, Length[csm[#]]==1]&], {ptn, IntegerPartitions[n-1]}]];
Table[Length[rurt[n]], {n, 10}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 06 2018
STATUS
approved