A316473 -id:A316473 - cp's OEIS frontend

A316501 Number of unlabeled rooted trees with n nodes in which the branches of any node with more than one distinct branch have empty intersection.

1, 1, 2, 4, 9, 19, 45, 103, 250, 611, 1528, 3853, 9875, 25481, 66382, 174085, 459541, 1219462

Offset: 1

Gus Wiseman, Jul 05 2018

			The a(6) = 19 rooted trees:
  (((((o)))))
  ((((oo))))
  (((o(o))))
  (((ooo)))
  ((o((o))))
  ((o(oo)))
  (((o)(o)))
  ((oo(o)))
  ((oooo))
  (o(((o))))
  (o((oo)))
  (o(o(o)))
  (o(ooo))
  ((o)((o)))
  (oo((o)))
  (oo(oo))
  (o(o)(o))
  (ooo(o))
  (ooooo)

Cf. A000081, A007562, A289509, A302796, A316470, A316473, A316475, A316500, A316502.

strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],Or[Length[Union[#]]==1,Intersection@@#=={}]&]];
Table[Length[strut[n]],{n,15}]

A317785 Number of locally connected rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 42, 55, 67, 91, 109, 144, 177, 228, 281, 366, 448, 579, 720, 916, 1142

Offset: 1

Gus Wiseman, Aug 06 2018

An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.

			The a(11) = 12 locally connected rooted trees:
  ((((((((((o))))))))))
  ((((((((o)(o))))))))
  (((((((o))((o)))))))
  ((((((o)))(((o))))))
  (((((o))))((((o)))))
  ((((((o)(o)(o))))))
  (((((o))((o)(o)))))
  ((((o))((o))((o))))
  ((((o)(o)(o)(o))))
  (((o))((o)(o)(o)))
  (((o)(o))((o)(o)))
  ((o)(o)(o)(o)(o))

Gus Wiseman, All 42 locally connected rooted trees with 16 nodes.

Cf. A000081, A276625, A286518, A286520, A301700, A304714, A316473, A316475, A317077, A317078, A317708, A317787.

multijoin[mss__]:=Join@@Table[Table[x, {Max[Count[#, x]&/@{mss}]}], {x, Union[mss]}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,Length[csm[#]]==1]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

A319271 Number of series-reduced locally non-intersecting aperiodic rooted trees with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 3, 9, 12, 27, 42, 91, 151, 312, 550, 1099, 2026, 3999, 7527, 14804, 28336, 55641, 107737, 211851, 413508, 814971, 1600512, 3162761, 6241234

Offset: 1

Gus Wiseman, Sep 16 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches, and aperiodic if the multiplicities in the multiset of branches directly under any given node are relatively prime, and locally non-intersecting if the branches directly under any given node with more than one branch have empty intersection.

			The a(8) = 9 rooted trees:
  (o(o(o(o))))
  (o(o(o)(o)))
  (o(ooo(o)))
  (oo(oo(o)))
  (o(o)(o(o)))
  (ooo(o(o)))
  (o(o)(o)(o))
  (ooo(o)(o))
  (ooooo(o))

Cf. A000081, A000837, A007562, A289509, A301700, A303431, A316470, A316473, A316475, A316495, A319270.

btrut[n_]:=btrut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[btrut/@c]]]/@IntegerPartitions[n-1],And[Intersection@@#=={},GCD@@Length/@Split[#]==1]&]];
Table[Length[btrut[n]],{n,30}]

A358460 Number of locally disjoint ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 103, 301, 902, 2767, 8637, 27324, 87409, 282319, 919352

Offset: 1

Gus Wiseman, Nov 19 2022

Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex.

			The a(1) = 1 through a(5) = 13 trees:
  o  (o)  (oo)   (ooo)    (oooo)
          ((o))  ((o)o)   ((o)oo)
                 ((oo))   ((oo)o)
                 (o(o))   ((ooo))
                 (((o)))  (o(o)o)
                          (o(oo))
                          (oo(o))
                          (((o))o)
                          (((o)o))
                          (((oo)))
                          ((o(o)))
                          (o((o)))
                          ((((o))))

The locally non-intersecting version is A143363, unordered A007562.
The unordered version is A316473, ranked by A316495.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A006013, A302696, A316471, A316694, A318185, A319378, A324768, A324844.

aot[n_]:=If[n==1,{{}},Join @@ Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,{_,{_,x_,_},_,{_,x_,_},_}]&]],{n,10}]

A319286 Number of series-reduced locally disjoint rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 2, 9, 67, 573, 6933, 97147, 1666999

Offset: 1

Gus Wiseman, Sep 16 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other branch of the same root.

			The a(3) = 9 trees:
  (1(11))
   (111)
  (1(12))
  (2(11))
   (112)
  (1(23))
  (2(13))
  (3(12))
   (123)
Examples of rooted trees that are not locally disjoint are ((11)(12)) and ((12)(13)).

Cf. A000081, A007562, A316473, A316475, A316494, A316495, A316496, A316497.

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]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]

A319285 Number of series-reduced locally stable rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 2, 9, 69, 619, 7739, 109855, 1898230

Offset: 1

Gus Wiseman, Sep 16 2018

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.

			The a(3) = 9 trees:
  (1(11))
   (111)
  (1(12))
  (2(11))
   (112)
  (1(23))
  (2(13))
  (3(12))
   (123)
Examples of rooted trees that are not locally stable are ((11)(111)), ((11)(112)), ((12)(112)), ((12)(123)).

Cf. A000081, A316466, A316468, A316469, A316473, A316475.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,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&])],stableQ]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]

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

Gus Wiseman, Sep 16 2018

			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}]

A319292 Number of series-reduced locally nonintersecting rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 1, 6, 48, 455, 5700, 83138, 1454870

Offset: 1

Gus Wiseman, Sep 16 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given node with at least two branches is empty.

			The a(3) = 6 trees are: (1(12)), (112), (1(23)), (2(13)), (3(12)), (123). Missing from this list but counted by A316651 are: (1(11)), (2(11)), (111).

Cf. A000081, A007562, A273873, A289509, A301700, A316470, A316473, A316475, A316651, A319271.

nonintQ[u_]:=Intersection@@u=={};
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&])],nonintQ]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]

cp's OEIS Frontend

A316501 Number of unlabeled rooted trees with n nodes in which the branches of any node with more than one distinct branch have empty intersection.

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A317785 Number of locally connected rooted trees with n nodes.

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A319271 Number of series-reduced locally non-intersecting aperiodic rooted trees with n nodes.

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A358460 Number of locally disjoint ordered rooted trees with n nodes.

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A319286 Number of series-reduced locally disjoint rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

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A319285 Number of series-reduced locally stable rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

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A319291 Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.

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A319292 Number of series-reduced locally nonintersecting rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

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