|
%I #14 Feb 07 2020 09:04:56
%S 1,1,2,3,6,13,28,62,143,338,804,1948,4789,11886,29796,75316,191702,
%T 491040,1264926,3274594,8514784,22229481,58243870
%N Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.
%C 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 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 The a(7) = 28 rooted trees:
%e 7,
%e (16),
%e (25),
%e (1(15)),
%e (34),
%e (1(24)), (2(14)), (4(12)), (124),
%e (1(1(14))),
%e (3(13)),
%e (2(23)),
%e (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), (12(13)), (13(12)),
%e (1(1(1(13)))),
%e (2(2(12))),
%e (1(1(2(12)))), (1(2(1(12)))), (1(12(12))), (2(1(1(12)))), (12(1(12))),
%e (1(1(1(1(12))))).
%e Missing from this list but counted by A300660 are ((12)(13)) and ((12)(1(12))).
%t disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
%t nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,disjointQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
%t Table[Length[nms[n]],{n,10}]
%Y Cf. A000081, A000669, A001678, A141268, A292504, A316653, A316654, A316656.
%Y The semi-identity tree version is A212804.
%Y Not requiring local disjointness gives A300660.
%Y The non-identity tree version is A316696.
%Y This is the case of A331686 where all leaves are singletons.
%Y Rooted identity trees are A004111.
%Y Locally disjoint rooted identity trees are A316471.
%Y Lone-child-avoiding locally disjoint rooted trees are A331680.
%Y Locally disjoint enriched identity p-trees are A331684.
%Y Cf. A306200, A316697, A331678, A331679, A331681, A331683, A331783, A331874.
%K nonn,more
%O 1,3
%A _Gus Wiseman_, Jul 10 2018
%E a(21)-a(23) from _Robert Price_, Sep 16 2018
%E Updated with corrected terminology by _Gus Wiseman_, Feb 06 2020
|
