cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Gus Wiseman, Nov 19 2022

Keywords

Comments

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

Examples

			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))))

Crossrefs

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.

Programs

  • Mathematica
    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}]

A317786 Matula-Goebel numbers of locally connected rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 9, 11, 23, 25, 27, 31, 81, 83, 97, 103, 115, 121, 125, 127, 243, 419, 431, 509, 515, 529, 563, 575, 625, 631, 661, 691, 709, 729, 961, 1067, 1331, 1543, 2095, 2187, 2369, 2575, 2645, 2875, 2897, 3001, 3125, 3637, 3691, 3803, 4091, 4201, 4637, 4663
Offset: 1

Views

Author

Gus Wiseman, Aug 07 2018

Keywords

Comments

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

Examples

			The sequence of locally connected trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   9: ((o)(o))
  11: ((((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  81: ((o)(o)(o)(o))
  83: ((((o)(o))))
  97: ((((o))((o))))

Crossrefs

A subset of A184155.
Cf. A000081, A276625, A286518, A286520, A304714, A316470, A316495, A316502, A317077, A317078, A317785.

Programs

  • Mathematica
    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]]]]]]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[Length[csm[primeMS/@primeMS[n]]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[1000],rupQ[#]&]

A317789 Matula-Goebel numbers of rooted trees that are not locally nonintersecting.

Original entry on oeis.org

9, 21, 23, 25, 27, 39, 46, 49, 57, 63, 65, 69, 73, 81, 83, 87, 91, 92, 97, 103, 111, 115, 117, 121, 125, 129, 133, 138, 146, 147, 159, 161, 166, 167, 169, 171, 183, 184, 185, 189, 194, 199, 203, 206, 207, 213, 219, 227, 230, 235, 237, 243, 247, 249, 253, 259
Offset: 1

Views

Author

Gus Wiseman, Aug 07 2018

Keywords

Comments

An unlabeled rooted tree is locally nonintersecting if there is no common subbranch to all branches directly under any given node.

Examples

			The sequence of rooted trees that are not locally nonintersecting together with their Matula-Goebel numbers begins:
   9: ((o)(o))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  39: ((o)(o(o)))
  46: (o((o)(o)))
  49: ((oo)(oo))
  57: ((o)(ooo))
  63: ((o)(o)(oo))
  65: (((o))(o(o)))
  69: ((o)((o)(o)))
  73: (((o)(oo)))
  81: ((o)(o)(o)(o))
  83: ((((o)(o))))
  87: ((o)(o((o))))
  91: ((oo)(o(o)))
  92: (oo((o)(o)))
  97: ((((o))((o))))

Crossrefs

Cf. A000081, A184155, A276625, A316470, A316495, A316502, A317785, A317786, A317787.

Programs

  • Mathematica
    rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],!rupQ[#]&]

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

Views

Author

Gus Wiseman, Sep 16 2018

Keywords

Comments

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.

Examples

			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)).

Crossrefs

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

Programs

  • Mathematica
    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}]

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

Views

Author

Gus Wiseman, Sep 16 2018

Keywords

Examples

			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))

Crossrefs

Cf. A000081, A007562, A301700, A316473, A316475, A316495, A316651, A316694, A316695, A316696, A316697, A319286.

Programs

  • Mathematica
    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}]
Previous Showing 21-25 of 25 results.

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