cp's OEIS Frontend

A319285 Number of series-reduced locally stable rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

%I A319285 #4 Sep 17 2018 08:34:34
%S A319285 1,2,9,69,619,7739,109855,1898230
%N A319285 Number of series-reduced locally stable rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
%C A319285 A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally stable if no branch is a submultiset of any other branch of the same root.
%e A319285 The a(3) = 9 trees:
%e A319285   (1(11))
%e A319285    (111)
%e A319285   (1(12))
%e A319285   (2(11))
%e A319285    (112)
%e A319285   (1(23))
%e A319285   (2(13))
%e A319285   (3(12))
%e A319285    (123)
%e A319285 Examples of rooted trees that are not locally stable are ((11)(111)), ((11)(112)), ((12)(112)), ((12)(123)).
%t A319285 submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t A319285 stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
%t A319285 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A319285 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A319285 gro[m_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],stableQ]];
%t A319285 Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]
%Y A319285 Cf. A000081, A316466, A316468, A316469, A316473, A316475.
%K A319285 nonn,more
%O A319285 1,2
%A A319285 _Gus Wiseman_, Sep 16 2018