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

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

Author

Gus Wiseman, Feb 01 2018

Keywords

• nonn

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

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

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.