

A316694


Number of lonechildavoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.


15



1, 1, 2, 3, 6, 13, 28, 62, 143, 338, 804, 1948, 4789, 11886, 29796, 75316, 191702, 491040, 1264926, 3274594, 8514784, 22229481, 58243870
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OFFSET

1,3


COMMENTS

A rooted tree is lonechildavoiding if every nonleaf 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.


LINKS

Table of n, a(n) for n=1..23.
Gus Wiseman, Sequences counting seriesreduced and lonechildavoiding trees by number of vertices.


EXAMPLE

The a(7) = 28 rooted trees:
7,
(16),
(25),
(1(15)),
(34),
(1(24)), (2(14)), (4(12)), (124),
(1(1(14))),
(3(13)),
(2(23)),
(1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), (12(13)), (13(12)),
(1(1(1(13)))),
(2(2(12))),
(1(1(2(12)))), (1(2(1(12)))), (1(12(12))), (2(1(1(12)))), (12(1(12))),
(1(1(1(1(12))))).
Missing from this list but counted by A300660 are ((12)(13)) and ((12)(1(12))).


MATHEMATICA

disjointQ[u_]:=Apply[And, Outer[#1==#2Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]], And[UnsameQ@@#, disjointQ[#]]&], {ptn, Rest[IntegerPartitions[n]]}], {n}];
Table[Length[nms[n]], {n, 10}]


CROSSREFS

Cf. A000081, A000669, A001678, A141268, A292504, A316653, A316654, A316656.
The semiidentity tree version is A212804.
Not requiring local disjointness gives A300660.
The nonidentity tree version is A316696.
This is the case of A331686 where all leaves are singletons.
Rooted identity trees are A004111.
Locally disjoint rooted identity trees are A316471.
Lonechildavoiding locally disjoint rooted trees are A331680.
Locally disjoint enriched identity ptrees are A331684.
Cf. A306200, A316697, A331678, A331679, A331681, A331683, A331783, A331874.
Sequence in context: A032143 A032160 A089735 * A000646 A316770 A197463
Adjacent sequences: A316691 A316692 A316693 * A316695 A316696 A316697


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Jul 10 2018


EXTENSIONS

a(21)a(23) from Robert Price, Sep 16 2018
Updated with corrected terminology by Gus Wiseman, Feb 06 2020


STATUS

approved



