A122827 Number of independent generators of degree n of the algebra of Free quasi-symmetric functions (or Malvenuto-Reutenauer algebra of permutations) as a dendriform dialgebra (i.e., number of totally primitive elements).

1, 0, 1, 6, 39, 284, 2305, 20682, 203651, 2186744, 25463925, 319989030, 4320183527, 62412737460, 961264517369, 15730347890082, 272650924761195, 4991218317261808, 96248879172426557, 1950405560049871134, 41440841509597888495, 921333064567137032620, 21392807067461981820417

Offset: 1

Author

Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Oct 23 2006

Keywords

• nonn

Comments

a(n) = (n-2)*A003319(n-1) for n >= 2 (result of Foissy). For instance 39 = 3 * 13 and 284 = 4 * 71. - F. Chapoton, Apr 26 2023

Links

• G. Duchamp, F. Hivert and J.-Y. Thibon, Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras, arXiv:math/0105065 [math.CO], 2001; Internat. J. Alg. Comp. 12 (2002), 671-717
• L. Foissy, Bidendriform bialgebras, trees and free quasi-symmetric functions, arXiv:math/0505207 [math.RA], 2005.
• L. Foissy, Plane posets, special posets, and permutations, Adv. Math. 240, 24-60 (2013).
• L. Foissy, Primitive elements of the Hopf algebra of free quasi-symmetric functions, Contemp. Math. 539, Amer. Math. Soc., 2011.
• Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions (2008); arXiv:0806.3682 [math.CO]; Discrete Math. 310 (2010), no. 24, 3584-3606.

Programs

Mathematica

terms = 23; f[t_] = 1 + Sum[n! t^n, {n, 1, terms+1}];
CoefficientList[(f[t]-1)/f[t]^2 + O[t]^(terms+1), t] // Rest (* Jean-François Alcover, Feb 13 2019 *)

Formula

G.f.: (f(t)-1)/f(t)^2, where f(t)=sum(n!*t^n,n>=0)

a(n) ~ n! * (1 - 4/n + 1/n^2 - 3/n^3 - 34/n^4 - 313/n^5 - 3189/n^6 - 36670/n^7 - 471381/n^8 - 6700559/n^9 - 104359132/n^10 - ...). - Vaclav Kotesovec, Feb 13 2019