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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).
2
1, 0, 1, 6, 39, 284, 2305, 20682, 203651, 2186744, 25463925, 319989030, 4320183527, 62412737460, 961264517369, 15730347890082, 272650924761195, 4991218317261808, 96248879172426557, 1950405560049871134, 41440841509597888495, 921333064567137032620, 21392807067461981820417
OFFSET
1,4
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.
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
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 *)
CROSSREFS
Sequence in context: A253077 A231482 A367233 * A103194 A009018 A289996
KEYWORD
nonn
AUTHOR
Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Oct 23 2006
STATUS
approved