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A316766 Number of series-reduced locally stable rooted identity trees whose leaves form an integer partition of n.

%I A316766 #8 Sep 15 2018 01:56:36
%S A316766 1,1,2,3,6,13,30,72,180,458,1194,3160,8459,22881,62417,171526,474405,
%T A316766 1319395,3687711,10352696,29178988
%N A316766 Number of series-reduced locally stable rooted identity trees whose leaves form an integer partition of n.
%C A316766 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. It is an identity tree if no branch appears multiple times under the same root.
%e A316766 The a(6) = 13 trees:
%e A316766 6,
%e A316766 (15),
%e A316766 (1(14)),
%e A316766 (1(1(13))),
%e A316766 (1(1(1(12)))),
%e A316766 (1(23)), (2(13)), (3(12)), (123),
%e A316766 (1(2(12))), (2(1(12))), (12(12)),
%e A316766 (24).
%e A316766 Example of non-stable trees are ((12)(123)) and ((12)(12(12))).
%t A316766 submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t A316766 stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
%t A316766 nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,stableQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
%t A316766 Table[Length[nms[n]],{n,10}]
%Y A316766 Cf. A000081, A000669, A001678, A004111, A141268, A292504, A300660, A316467, A316474, A316653, A316654, A316656.
%K A316766 nonn,more
%O A316766 1,3
%A A316766 _Gus Wiseman_, Jul 12 2018
%E A316766 a(18)-a(21) from _Robert Price_, Sep 14 2018