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A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.

%I A316694 #14 Feb 07 2020 09:04:56
%S A316694 1,1,2,3,6,13,28,62,143,338,804,1948,4789,11886,29796,75316,191702,
%T A316694 491040,1264926,3274594,8514784,22229481,58243870
%N A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.
%C A316694 A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root. It is an identity tree if no branch appears multiple times under the same root.
%H A316694 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>
%e A316694 The a(7) = 28 rooted trees:
%e A316694   7,
%e A316694   (16),
%e A316694   (25),
%e A316694   (1(15)),
%e A316694   (34),
%e A316694   (1(24)), (2(14)), (4(12)), (124),
%e A316694   (1(1(14))),
%e A316694   (3(13)),
%e A316694   (2(23)),
%e A316694   (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), (12(13)), (13(12)),
%e A316694   (1(1(1(13)))),
%e A316694   (2(2(12))),
%e A316694   (1(1(2(12)))), (1(2(1(12)))), (1(12(12))), (2(1(1(12)))), (12(1(12))),
%e A316694   (1(1(1(1(12))))).
%e A316694 Missing from this list but counted by A300660 are ((12)(13)) and ((12)(1(12))).
%t A316694 disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t A316694 nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,disjointQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
%t A316694 Table[Length[nms[n]],{n,10}]
%Y A316694 Cf. A000081, A000669, A001678, A141268, A292504, A316653, A316654, A316656.
%Y A316694 The semi-identity tree version is A212804.
%Y A316694 Not requiring local disjointness gives A300660.
%Y A316694 The non-identity tree version is A316696.
%Y A316694 This is the case of A331686 where all leaves are singletons.
%Y A316694 Rooted identity trees are A004111.
%Y A316694 Locally disjoint rooted identity trees are A316471.
%Y A316694 Lone-child-avoiding locally disjoint rooted trees are A331680.
%Y A316694 Locally disjoint enriched identity p-trees are A331684.
%Y A316694 Cf. A306200, A316697, A331678, A331679, A331681, A331683, A331783, A331874.
%K A316694 nonn,more
%O A316694 1,3
%A A316694 _Gus Wiseman_, Jul 10 2018
%E A316694 a(21)-a(23) from _Robert Price_, Sep 16 2018
%E A316694 Updated with corrected terminology by _Gus Wiseman_, Feb 06 2020