cp's OEIS Frontend

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).

%I A122827 #31 May 20 2023 15:06:17
%S A122827 1,0,1,6,39,284,2305,20682,203651,2186744,25463925,319989030,
%T A122827 4320183527,62412737460,961264517369,15730347890082,272650924761195,
%U A122827 4991218317261808,96248879172426557,1950405560049871134,41440841509597888495,921333064567137032620,21392807067461981820417
%N 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).
%C A122827 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
%H A122827 G. Duchamp, F. Hivert and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0105065">Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras</a>, arXiv:math/0105065 [math.CO], 2001; Internat. J. Alg. Comp. 12 (2002), 671-717
%H A122827 L. Foissy, <a href="https://arxiv.org/abs/math/0505207">Bidendriform bialgebras, trees and free quasi-symmetric functions</a>, arXiv:math/0505207 [math.RA], 2005.
%H A122827 L. Foissy, <a href="https://doi.org/10.1016/j.aim.2013.03.007">Plane posets, special posets, and permutations</a>,  Adv. Math. 240, 24-60 (2013).
%H A122827 L. Foissy, <a href="https://doi.org/10.1090/conm/539/10629">Primitive elements of the Hopf algebra of free quasi-symmetric functions</a>, Contemp. Math. 539, Amer. Math. Soc., 2011.
%H A122827 Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/0806.3682">Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions</a> (2008); arXiv:0806.3682 [math.CO]; Discrete Math. 310 (2010), no. 24, 3584-3606.
%F A122827 G.f.: (f(t)-1)/f(t)^2, where f(t)=sum(n!*t^n,n>=0)
%F A122827 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
%t A122827 terms = 23; f[t_] = 1 + Sum[n! t^n, {n, 1, terms+1}];
%t A122827 CoefficientList[(f[t]-1)/f[t]^2 + O[t]^(terms+1), t] // Rest (* _Jean-François Alcover_, Feb 13 2019 *)
%K A122827 nonn
%O A122827 1,4
%A A122827 Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Oct 23 2006