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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A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 32, 36, 38, 42, 48, 54, 56, 64, 72, 76, 78, 84, 96, 98, 106, 108, 112, 114, 126, 128, 144, 152, 156, 162, 168, 192, 196, 212, 216, 222, 224, 228, 234, 252, 256, 262, 266, 288, 294, 304, 312, 318, 324, 336, 342, 366, 378
Offset: 1

Views

Author

Gus Wiseman, Aug 20 2018

Keywords

Comments

A number x is totally transitive if (1) whenever prime(y) divides x it follows that y is totally transitive and (2) if prime(y) divides x and prime(z) divides y then prime(z) also divides x.

Examples

			The sequence of all totally transitive rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  76: (oo(ooo))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))
  98: (o(oo)(oo))

Crossrefs

Cf. A000081, A001678, A004111, A007097, A061775, A276625, A279861, A290689, A290760, A290822, A291636, A318185, A318187.

Programs

  • Mathematica
    subprimes[n_]:=If[n==1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
trmgQ[n_]:=Or[n==1,And[Divisible[n,Times@@subprimes[n]],And@@Cases[FactorInteger[n],{p_,_}:>trmgQ[PrimePi[p]]]]];
Select[Range[100],trmgQ]

A318227 Number of inequivalent leaf-colorings of rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 3, 5, 14, 38, 114, 330, 1054, 3483, 11841, 41543, 149520, 552356, 2084896, 8046146, 31649992, 127031001, 518434863, 2153133594, 9081863859, 38909868272, 169096646271, 745348155211, 3329032020048, 15063018195100, 68998386313333, 319872246921326, 1500013368166112
Offset: 1

Views

Author

Gus Wiseman, Aug 21 2018

Keywords

Comments

In a rooted identity tree, all branches directly under any given branch are different.
The leaves are colored after selection of the tree. Since all trees are asymmetric, the symmetry group of the leaves will be the identity group and a tree with k leaves will have Bell(k) inequivalent leaf-colorings. - Andrew Howroyd, Dec 10 2020

Examples

			Inequivalent representatives of the a(6) = 14 leaf-colorings:
  (1(1(1)))  ((1)((1)))  (1(((1))))  ((1((1))))  (((1(1))))  (((((1)))))
  (1(1(2)))  ((1)((2)))  (1(((2))))  ((1((2))))  (((1(2))))
  (1(2(1)))
  (1(2(2)))
  (1(2(3)))

Links

Crossrefs

Cf. A000081, A001190, A001678, A003238, A004111, A290689, A318185, A304486.
Cf. A318226, A318228, A318229, A318230, A318231, A318234.
Cf. A000110 (Bell numbers), A055327, A301342.

Programs

  • Mathematica
    idt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[idt/@c]],UnsameQ@@#&],{c,IntegerPartitions[n-1]}]];
Table[Sum[BellB[Count[tree,{},{0,Infinity}]],{tree,idt[n]}],{n,16}]
  • PARI
    \\ bell(n) is A000110(n).
WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
bell(n)={sum(k=1, n, stirling(n,k,2))}
seq(n)={my(v=[y], b=vector(n,k,bell(k))); for(n=2, n, v=concat(v[1], WeighMT(v))); vector(n, k, sum(i=1, k, polcoef(v[k],i)*b[i]))} \\ Andrew Howroyd, Dec 10 2020

Formula

a(n) = Sum_{k=1..n} A055327(n,k) * A000110(k). - Andrew Howroyd, Dec 10 2020

Extensions

Terms a(17) and beyond from Andrew Howroyd, Dec 10 2020

A318226 Number of inequivalent leaf-colorings of rooted trees with n nodes.

Original entry on oeis.org

1, 1, 3, 8, 25, 80, 286, 1070, 4280, 17946, 78907, 361248, 1718001, 8456130, 42980034, 225066289, 1212028798, 6701265897, 37986122037, 220477639797, 1308833637621, 7938564964369, 49151551028767, 310388888456536, 1997635594602629, 13093695854320203, 87349973125826943
Offset: 1

Views

Author

Gus Wiseman, Aug 21 2018

Keywords

Examples

			Inequivalent representatives of the a(5) = 25 leaf-colorings:
(1111) (11(1)) (1(11)) ((111)) ((1)(1)) (1((1))) ((1(1))) (((11))) ((((1))))
(1112) (11(2)) (1(12)) ((112)) ((1)(2)) (1((2))) ((1(2))) (((12)))
(1122) (12(1)) (1(22)) ((123))
(1123) (12(3)) (1(23))
(1234)

Crossrefs

Cf. A000081, A001190, A001678, A003238, A004111, A290689, A318185, A304486.
Cf. A318227, A318228, A318229, A318230, A318231, A318234, A339645.

Programs

  • Mathematica
    undats[m_]:=Union[DeleteCases[Cases[m,_?AtomQ,{0,Infinity},Heads->True],List]];
expnorm[m_]:=If[Length[undats[m]]==0,m,If[undats[m]!=Range[Max@@undats[m]],expnorm[m/.Rule@@@Table[{(undats[m])[[i]],i},{i,Length[undats[m]]}]],First[Sort[expnorm[m,1]]]]];expnorm[m_,aft_]:=If[Length[undats[m]]<=aft,{m},With[{mx=Table[Count[m,i,{0,Infinity},Heads->True],{i,Select[undats[m],#>=aft&]}]},Union@@(expnorm[#,aft+1]&/@Union[Table[MapAt[Sort,m/.{par+aft-1->aft,aft->par+aft-1},Position[m,[__]]],{par,First/@Position[mx,Max[mx]]}]])]];
urt[n_]:=urt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[urt/@c]],{c,IntegerPartitions[n-1]}]];
slip[e_,l_,q_]:=ReplacePart[e,Rule@@@Transpose[{Position[e,l],q}]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Join@@Table[Union[expnorm/@Table[slip[tree,{},seq],{seq,Join@@Permutations/@allnorm[Count[tree,{},{0,Infinity},Heads->True]]}]],{tree,urt[n]}]],{n,7}]
  • PARI
    \\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(Z=x*sv(1), p = Z + O(x^2)); for(n=2, n, p = Z-x + x*sEulerT(p)); p}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 13 2020

Extensions

Terms a(9) and beyond from Andrew Howroyd, Dec 10 2020

A318229 Number of inequivalent leaf-colorings of transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 92, 255
Offset: 1

Views

Author

Gus Wiseman, Aug 21 2018

Keywords

Comments

In a transitive rooted tree, every branch of a branch of the root is also a branch of the root.

Examples

			Inequivalent representatives of the a(5) = 13 leaf-colorings:
  (1111)  (1(11))  (11(1))
  (1112)  (1(12))  (11(2))
  (1122)  (1(22))  (12(1))
  (1123)  (1(23))  (12(3))
  (1234)

Crossrefs

Cf. A000081, A001190, A001678, A003238, A004111, A279861, A290689, A290822, A318185, A304486.
Cf. A318226, A318227, A318228, A318230, A318231, A318234.

A318234 Number of inequivalent leaf-colorings of totally transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 87
Offset: 1

Views

Author

Gus Wiseman, Aug 21 2018

Keywords

Comments

A rooted tree is totally transitive if every branch of the root is totally transitive and every branch of a branch of the root is also a branch of the root.

Examples

			Inequivalent representatives of the a(6) = 34 leaf-colorings:
  (11(11))  (11111)  (111(1))  (1(111))  (1(1)(1))
  (11(12))  (11112)  (111(2))  (1(112))  (1(1)(2))
  (11(22))  (11122)  (112(1))  (1(122))  (1(2)(2))
  (11(23))  (11123)  (112(2))  (1(123))  (1(2)(3))
  (12(11))  (11223)  (112(3))  (1(222))
  (12(12))  (11234)  (123(1))  (1(223))
  (12(13))  (12345)  (123(4))  (1(234))
  (12(33))
  (12(34))

Crossrefs

Cf. A000081, A001190, A001678, A003238, A004111, A290689, A318185, A304486, A318186, A318187.
Cf. A318226, A318227, A318228, A318229, A318230, A318231.

A358456 Number of recursively bi-anti-transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 7, 17, 47, 117, 321, 895, 2556, 7331, 21435, 63116, 187530
Offset: 1

Views

Author

Gus Wiseman, Nov 18 2022

Keywords

Comments

We define an unlabeled ordered rooted tree to be recursively bi-anti-transitive if there are no two branches of the same node such that one is a branch of the other.

Examples

			The a(1) = 1 through a(6) = 17 trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((o))o)   (((o))oo)
                          (((oo)))   (((oo))o)
                          ((o)(o))   (((ooo)))
                          (o((o)))   ((o)(oo))
                          ((((o))))  ((oo)(o))
                                     (o((o))o)
                                     (o((oo)))
                                     (oo((o)))
                                     ((((o)))o)
                                     ((((o))o))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Crossrefs

The unordered version is A324765, ranked by A324766.
The directed version is A358455.
A000108 counts ordered rooted trees, unordered A000081.
A306844 counts anti-transitive rooted trees.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A318185, A324695, A324751, A324756, A324758, A324764, A324767, A324768, A324838, A324840, 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_,_},_}|{_,{_,x_,_},_,x_,_}]&]],{n,10}]

A318187 Number of totally transitive rooted trees with n leaves.

Original entry on oeis.org

2, 2, 4, 8, 16, 32, 62, 122, 234, 451, 857, 1630, 3068, 5772, 10778, 20093, 37259
Offset: 1

Views

Author

Gus Wiseman, Aug 20 2018

Keywords

Comments

A rooted tree is totally transitive if every branch of the root is totally transitive and every branch of a branch of the root is also a branch of the root.

Examples

			The a(5) = 16 totally transitive rooted trees with 5 leaves:
  (o(o)(o(o)(o)))
  (o(o)(o)(o(o)))
  (o(o)(o)(o)(o))
  (o(o)(oo(o)))
  (oo(o)(o(o)))
  (o(o)(o)(oo))
  (oo(o)(o)(o))
  (o(o)(ooo))
  (o(oo)(oo))
  (oo(o)(oo))
  (ooo(o)(o))
  (o(oooo))
  (oo(ooo))
  (ooo(oo))
  (oooo(o))
  (ooooo)

Crossrefs

Cf. A000081, A000669, A001678, A004111, A050381, A279861, A290689, A290760, A290822, A318185, A318186.

Programs

  • Mathematica
    totralv[n_]:=totralv[n]=If[n==1,{{},{{}}},Join@@Table[Select[Union[Sort/@Tuples[totralv/@c]],Complement[Union@@#,#]=={}&],{c,Select[IntegerPartitions[n],Length[#]>1&]}]];
Table[Length[totralv[n]],{n,8}]

A358454 Number of weakly transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 33, 80, 201, 509, 1330, 3432, 8982, 23559, 62189
Offset: 1

Views

Author

Gus Wiseman, Nov 18 2022

Keywords

Comments

We define an unlabeled ordered rooted tree to be weakly transitive if every branch of a branch of the root is itself a branch of the root.

Examples

			The a(1) = 1 through a(6) = 13 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)
                ((o)o)  ((o)oo)  ((o)ooo)
                (o(o))  ((oo)o)  ((oo)oo)
                        (o(o)o)  ((ooo)o)
                        (o(oo))  (o(o)oo)
                        (oo(o))  (o(oo)o)
                                 (o(ooo))
                                 (oo(o)o)
                                 (oo(oo))
                                 (ooo(o))
                                 ((o)(o)o)
                                 ((o)o(o))
                                 (o(o)(o))

Crossrefs

The unordered version is A290689, ranked by A290822.
The directed version is A358453.
A000081 counts rooted trees.
A306844 counts anti-transitive rooted trees.
Cf. A318185, A324695, A324751, A324756, A324758, A324764-A324768, A324838, A324840, A324844, A358456.

Programs

  • Mathematica
    aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Complement[Union@@#,#]=={}&]],{n,10}]

A358455 Number of recursively anti-transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 72, 206, 608, 1830, 5612, 17442, 54866, 174252, 558072, 1800098
Offset: 1

Views

Author

Gus Wiseman, Nov 18 2022

Keywords

Comments

We define an unlabeled ordered rooted tree to be recursively anti-transitive if no branch of a branch of a subtree is a branch of the same subtree farther to the left.

Examples

			The a(1) = 1 through a(5) = 10 trees:
  o  (o)  (oo)   (ooo)    (oooo)
          ((o))  ((o)o)   ((o)oo)
                 ((oo))   ((oo)o)
                 (((o)))  ((ooo))
                          (((o))o)
                          (((o)o))
                          (((oo)))
                          ((o)(o))
                          (o((o)))
                          ((((o))))

Crossrefs

The unordered version is A324765, ranked by A324766.
The undirected version is A358456.
A000108 counts ordered rooted trees, unordered A000081.
A306844 counts anti-transitive rooted trees.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A318185, A324695, A324751, A324756, A324758, A324764, A324767, A324768, A324838, A324840, 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}]

A324841 Matula-Goebel numbers of fully recursively anti-transitive rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 21, 23, 25, 27, 31, 32, 35, 49, 51, 53, 57, 59, 63, 64, 67, 73, 77, 81, 83, 85, 95, 97, 103, 115, 121, 125, 127, 128, 131, 133, 147, 149, 153, 159, 161, 171, 175, 177, 187, 189, 201, 209, 217, 227, 233, 241, 243, 245
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.

Examples

			The sequence of fully recursively anti-transitive rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  32: (ooooo)
  35: (((o))(oo))
  49: ((oo)(oo))
  51: ((o)((oo)))
  53: ((oooo))
  57: ((o)(ooo))
  59: ((((oo))))
  63: ((o)(o)(oo))

Crossrefs

Cf. A000081, A007097, A290689, A303431, A304360, A306844, A316502, A318185, A318186.
Cf. A324695, A324751, A324756, A324758, A324766, A324768, A324769.
Cf. A324838, A324840, A324844.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fratQ[n_]:=And[Intersection[Union@@Rest[FixedPointList[Union@@primeMS/@#&,primeMS[n]]],primeMS[n]]=={},And@@fratQ/@primeMS[n]];
Select[Range[100],fratQ]
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