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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A299024 Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors.

Original entry on oeis.org

1, 1, 3, 4, 7, 13, 21, 34, 58, 98, 158, 258, 421, 676, 1108, 1777, 2836, 4544, 7220, 11443, 18215, 28729, 45203, 71139, 111518, 174402, 272367, 424892, 660563, 1025717, 1590448, 2460346, 3800816, 5862640, 9026963, 13885425, 21321663, 32695098, 50073855
Offset: 1

Views

Author

Gus Wiseman, Jan 31 2018

Keywords

Examples

			The a(5) = 7 compositions:
      (5) = (5)
     (41) = (4)*(1)
     (14) = (14)
     (32) = (3)*(2)
     (23) = (23)
    (131) = (13)*(1)
    (212) = (2)*(12)
Not included:
    (311) = (3)*(1)*(1)
    (113) = (113)
    (221) = (2)*(2)*(1)
    (122) = (122)
   (2111) = (2)*(1)*(1)*(1)
   (1211) = (12)*(1)*(1)
   (1121) = (112)*(1)
   (1112) = (1112)
  (11111) = (1)*(1)*(1)*(1)*(1)

Links

Crossrefs

Cf. A001045, A032020, A032153, A034691, A049311, A050342, A059966, A089259, A098407, A116540, A185700, A270995, A283877, A292432, A296373, A299023, A299026, A299027.

Programs

  • Mathematica
    nn=50;
ser=Product[(1+x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(N)={WeighT(Vec(sum(n=1, N-1, (n-1)!*x^(n*(n+1)/2)/prod(k=1, n, 1-x^k + O(x^N)))))} \\ Andrew Howroyd, Dec 01 2018

Formula

Weigh transform of A032153.

A299026 Number of compositions of n whose standard factorization into Lyndon words has all weakly increasing factors.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 59, 111, 205, 378, 685, 1238, 2213, 3940, 6955, 12221, 21333, 37074, 64073, 110267, 188877, 322244, 547522, 926903, 1563370, 2628008, 4402927, 7353656, 12244434, 20329271, 33657560, 55574996, 91525882, 150356718, 246403694, 402861907
Offset: 1

Views

Author

Gus Wiseman, Feb 01 2018

Keywords

Examples

			The 2^6 - a(7) = 5 compositions of 7 whose Lyndon prime factors are not all weakly increasing: (11212), (1132), (1213), (1321), (142).

Links

Crossrefs

Cf. A001045, A032153, A034691, A049311, A059966, A098407, A167934, A185700, A270995, A296373, A299023, A299024, A299027.

Programs

  • Mathematica
    nn=50;
ser=Product[1/(1-x^n)^(PartitionsP[n]-DivisorSigma[0,n]+1),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={EulerT(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ Andrew Howroyd, Dec 01 2018

Formula

Euler transform of A167934.

A299027 Number of compositions of n whose standard factorization into Lyndon words has all distinct weakly increasing factors.

Original entry on oeis.org

1, 1, 3, 5, 11, 20, 38, 69, 125, 225, 400, 708, 1244, 2176, 3779, 6532, 11229, 19223, 32745, 55555, 93875, 158025, 265038, 443009, 738026, 1225649, 2029305, 3350167, 5515384, 9055678, 14830076, 24226115, 39480306, 64190026, 104130753, 168556588, 272268482
Offset: 1

Views

Author

Gus Wiseman, Feb 01 2018

Keywords

Examples

			The a(5) = 11 compositions:
      (5) = (5)
     (41) = (4)*(1)
     (14) = (14)
     (32) = (3)*(2)
     (23) = (23)
    (131) = (13)*(1)
    (113) = (113)
    (212) = (2)*(12)
    (122) = (122)
   (1121) = (112)*(1)
   (1112) = (1112)
Not included:
    (311) = (3)*(1)*(1)
    (221) = (2)*(2)*(1)
   (2111) = (2)*(1)*(1)*(1)
   (1211) = (12)*(1)*(1)
  (11111) = (1)*(1)*(1)*(1)*(1)

Links

Crossrefs

Cf. A001045, A032153, A034691, A049311, A059966, A098407, A116540, A185700, A270995, A283877, A292432, A293993, A296373, A299023, A299024, A299026.

Programs

  • Mathematica
    nn=50;
ser=Product[(1+x^n)^(PartitionsP[n]-DivisorSigma[0,n]+1),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={WeighT(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ Andrew Howroyd, Dec 01 2018

Formula

Weigh transform of A167934.

A299072 Sequence is an irregular triangle read by rows with zeros removed where T(n,k) is the number of compositions of n whose standard factorization into Lyndon words has k distinct factors.

Original entry on oeis.org

1, 2, 3, 1, 5, 3, 7, 9, 13, 17, 2, 19, 39, 6, 35, 72, 21, 59, 141, 55, 1, 107, 266, 132, 7, 187, 511, 300, 26, 351, 952, 660, 85, 631, 1827, 1395, 240, 3, 1181, 3459, 2901, 636, 15, 2191, 6595, 5977, 1554, 67, 4115, 12604, 12123, 3698, 228, 7711, 24173, 24504
Offset: 1

Views

Author

Gus Wiseman, Feb 01 2018

Keywords

Comments

Row sums are 2^(n-1). First column is A008965. A regular version is A299070.

Examples

			Triangle begins:
    1
    2
    3    1
    5    3
    7    9
   13   17    2
   19   39    6
   35   72   21
   59  141   55    1
  107  266  132    7
  187  511  300   26

Links

Crossrefs

Cf. A001045, A001221, A008965, A059966, A116608, A146289, A185700, A296373, A299070.

Programs

  • Mathematica
    LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
qit[q_]:=If[#===Length[q],{q},Prepend[qit[Drop[q,#]],Take[q,#]]]&[Max@@Select[Range[Length[q]],LyndonQ[Take[q,#]]&]];
DeleteCases[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Union[qit[#]]]===k&]],{n,11},{k,n}],0,{2}]
  • PARI
    \\ here b(n) is A059966.
b(n)={sumdiv(n, d, moebius(n/d) * (2^d-1))/n}
A(n)=[Vecrev(p/y) | p<-Vec(prod(k=1, n, (1 - y + y/(1-x^k) + O(x*x^n))^b(k))-1)]
my(T=A(15)); for(n=1, #T, print(T[n])) \\ Andrew Howroyd, Dec 08 2018

A299070 Regular triangle T(n,k) is the number of compositions of n whose standard factorization into Lyndon words has k distinct factors.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 5, 3, 0, 0, 7, 9, 0, 0, 0, 13, 17, 2, 0, 0, 0, 19, 39, 6, 0, 0, 0, 0, 35, 72, 21, 0, 0, 0, 0, 0, 59, 141, 55, 1, 0, 0, 0, 0, 0, 107, 266, 132, 7, 0, 0, 0, 0, 0, 0, 187, 511, 300, 26, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Feb 01 2018

Keywords

Comments

Row sums are 2^(n-1). First column is A008965. A version without the zeros is A299072.

Examples

			Triangle begins:
    1
    2    0
    3    1    0
    5    3    0    0
    7    9    0    0    0
   13   17    2    0    0    0
   19   39    6    0    0    0    0
   35   72   21    0    0    0    0    0
   59  141   55    1    0    0    0    0    0
  107  266  132    7    0    0    0    0    0    0
  187  511  300   26    0    0    0    0    0    0    0.
The a(5,2) = 9 compositions are (41), (32), (311), (131), (221), (212), (2111), (1211), (1121) with factorizations
    (41) = (4) * (1)
    (32) = (3) * (2)
   (311) = (3) * (1)^2
   (131) = (13) * (1)
   (221) = (2)^2 * (1)
   (212) = (2) * (12)
  (2111) = (2) * (1)^3
  (1211) = (12) * (1)^2
  (1121) = (112) * (1).

Crossrefs

Cf. A001045, A001221, A008965, A059966, A116608, A146289, A185700, A296373, A299072.

Programs

  • Mathematica
    LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
qit[q_]:=If[#===Length[q],{q},Prepend[qit[Drop[q,#]],Take[q,#]]]&[Max@@Select[Range[Length[q]],LyndonQ[Take[q,#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Union[qit[#]]]===k&]],{n,11},{k,n}]
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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