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.

Showing 1-7 of 7 results.

A318231 Number of inequivalent leaf-colorings of series-reduced rooted trees with n nodes.

Original entry on oeis.org

1, 0, 2, 3, 9, 23, 73, 229, 796, 2891, 11118, 44695, 187825, 820320, 3716501, 17413308, 84209071, 419461933, 2148673503, 11301526295, 60956491070, 336744177291, 1903317319015, 10995856040076, 64873456288903, 390544727861462, 2397255454976268, 14993279955728851
Offset: 1

Views

Author

Gus Wiseman, Aug 21 2018

Keywords

Comments

In a series-reduced rooted tree, every non-leaf node has at least two branches.

Examples

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

Crossrefs

Cf. A000081, A001190, A001678, A003238, A004111, A290689, A291636, A304486.
Cf. A318226, A318227, A318228, A318229, A318230, A318234, A339645, A339648.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(concat(v[1..n-2], [0]))), n-1 )); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Extensions

Terms a(8) and beyond from Andrew Howroyd, Dec 11 2020

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

A318230 Number of inequivalent leaf-colorings of binary rooted trees with 2n + 1 nodes.

Original entry on oeis.org

1, 2, 4, 18, 79, 474, 3166, 24451, 208702, 1958407, 19919811, 217977667, 2547895961, 31638057367, 415388265571, 5743721766718, 83356613617031, 1265900592208029, 20064711719120846, 331153885800672577, 5679210649417608867, 101017359002718628295, 1860460510677429522171
Offset: 0

Views

Author

Gus Wiseman, Aug 21 2018

Keywords

Examples

			Inequivalent representatives of the a(3) = 18 leaf-colorings of binary rooted trees with 7 nodes:
  (1(1(11)))  ((11)(11))
  (1(1(12)))  ((11)(12))
  (1(1(22)))  ((11)(22))
  (1(1(23)))  ((11)(23))
  (1(2(11)))  ((12)(12))
  (1(2(12)))  ((12)(13))
  (1(2(13)))  ((12)(34))
  (1(2(22)))
  (1(2(23)))
  (1(2(33)))
  (1(2(34)))

Links

Crossrefs

Cf. A000081, A001190, A001678, A003238, A004111, A111299, A290689, A304486.
Cf. A318226, A318227, A318228, A318229, A318231, A318234, A319590, A339645.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, my(p=x*Ser(v[1..n-1])); v[n]=polcoef(p^2 + if(n%2==0, sRaise(p,2)), n)/2); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(20)) \\ Andrew Howroyd, Dec 11 2020

Extensions

Terms a(5) 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

A318228 Number of inequivalent leaf-colorings of planted achiral trees with n nodes.

Original entry on oeis.org

1, 1, 3, 6, 13, 20, 43, 58, 115, 171, 323, 379, 1034, 1135, 2321, 4327, 8915, 9212, 33939, 34429, 128414, 234017, 417721, 418976, 2931624, 5096391, 11770830, 20357876, 64853630, 64858195
Offset: 1

Views

Author

Gus Wiseman, Aug 21 2018

Keywords

Comments

In a planted achiral tree, all branches directly under any given branch are identical.

Examples

			Inequivalent representatives of the a(5) = 13 leaf-colorings:
  (1111)  ((111))  ((1)(1))  (((11)))  ((((1))))
  (1112)  ((112))  ((1)(2))  (((12)))
  (1122)  ((123))
  (1123)
  (1234)

Crossrefs

Cf. A000081, A001190, A001678, A003238, A004111, A214577, A290689, A304486.
Cf. A318226, A318227, A318229, A318230, A318231, A318234, A339645.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
G(v)={my(t=2, p=sv(1)); for(i=1, #v, my(d=v[i]); if(d>1, p=sApplyCI(symGroupCycleIndex(d), d, p, t)); t=t*d+1); p}
cycleIndex(n)={my(recurse(r,v)=if(r==1, G(v), sumdiv(r-1, d, self()((r-1)/d, concat(d,v))))); recurse(n,[])}
a(n)={StructsByCycleIndex(n, cycleIndex(n), n)} \\ Andrew Howroyd, Dec 13 2020

Extensions

a(9)-a(30) from Andrew Howroyd, Dec 11 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.
Showing 1-7 of 7 results.

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