A331684
Number of locally disjoint enriched identity p-trees of weight n.
Original entry on oeis.org
1, 1, 2, 3, 6, 14, 30, 68, 157, 379, 901, 2229, 5488, 13846, 34801, 89368, 228186, 592943, 1533511, 4026833
Offset: 1
The a(1) = 1 through a(6) = 14 enriched p-trees:
1 2 3 4 5 6
(21) (31) (32) (42)
((21)1) (41) (51)
((21)2) (321)
((31)1) ((21)3)
(((21)1)1) ((31)2)
((32)1)
(3(21))
((41)1)
((21)21)
(((21)1)2)
(((21)2)1)
(((31)1)1)
((((21)1)1)1)
The non-identity version is
A331687.
Locally disjoint identity trees are
A316471.
Enriched identity p-trees are
A331875, with locally disjoint case
A331687.
Cf.
A000669,
A005804,
A141268,
A300660,
A316696,
A316697,
A331678,
A331679,
A331680,
A331683,
A331686,
A331783,
A331874.
-
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldeip[n_]:=Prepend[Select[Join@@Table[Tuples[ldeip/@p],{p,Rest[IntegerPartitions[n]]}],UnsameQ@@#&&disjointQ[DeleteCases[#,_Integer]]&],n];
Table[Length[ldeip[n]],{n,12}]
A316770
Number of series-reduced locally nonintersecting rooted identity trees whose leaves form an integer partition of n.
Original entry on oeis.org
1, 1, 2, 3, 6, 13, 28, 64, 153, 379, 939, 2385, 6121, 15871, 41529, 109509, 290607, 775842, 2081874, 5612176, 15191329, 41274052
Offset: 1
The a(6) = 13 trees:
(1(1(1(12))))
(1(1(13)))
(1(2(12)))
(2(1(12)))
(12(12))
(1(14))
(1(23))
(2(13))
(3(12))
(123)
(15)
(24)
6
Examples of series-reduced rooted identity trees that are not locally nonintersecting are ((12)(13)) and ((12)(1(12))).
Cf.
A000081,
A000669,
A001678,
A141268,
A292504,
A316500,
A316651,
A316652,
A316655,
A316696,
A316768.
-
nonintQ[u_]:=Intersection@@u=={};
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,nonintQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,15}]
A319291
Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.
Original entry on oeis.org
1, 2, 12, 107, 1299, 20764, 412957, 9817743
Offset: 1
The a(3) = 12 series-reduced locally disjoint rooted trees:
(1(11))
(111)
(1(22))
(2(12))
(122)
(1(12))
(2(11))
(112)
(1(23))
(2(13))
(3(12))
(123)
The trees counted by A316651(4) but not by a(4):
((11)(12))
((12)(13))
((12)(22))
((12)(23))
((13)(23))
Cf.
A000081,
A007562,
A301700,
A316473,
A316475,
A316495,
A316651,
A316694,
A316695,
A316696,
A316697,
A319286.
-
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],disjointQ]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]
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