N. Bourbaki, Groupes et algèbres de lie, 1968.

P. Cartier, Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents A primer of hopf algebras, Frontiers in number theory, physics, and geometry II, no. 9. 3, pp.137-173, 1972.

F. Chapoton, Alg??bres de Hopf des permutah??dres, associah??dres et hypercubes, Opérades différentielles graduées sur les simplexes et lespermutò edres, Bulletin de la Société mathématique de France, pp.264-275, 2000.
DOI : 10.1006/aima.1999.1868

K. Ebrahimi-fard, S. J. Malham, F. Patras, and A. Wiese, Flows and stochastic Taylor series in It?? calculus, Journal of Physics A: Mathematical and Theoretical, vol.48, issue.49, p.495202, 2015.
DOI : 10.1088/1751-8113/48/49/495202

K. Ebrahimi-fard, S. J. Malham, F. Patras, and A. Wiese, The exponential Lie series for continuous semimartingales, Proc. R. Soc. A, p.20150429, 2015.
DOI : 10.1098/rspa.2009.0203
URL : https://hal.archives-ouvertes.fr/hal-01287021

S. Eilenberg and S. M. Lane, On the Groups H(??, n), I, The Annals of Mathematics, vol.58, issue.1, pp.55-106, 1953.
DOI : 10.2307/1969820

F. Fauvet and F. Menous, Ecalle's arborification-coarborification transforms and connes-kreimer hopf algebra, arXiv preprint arXiv:1212, p.4740, 2012.

L. Foissy, Bidendriform bialgebras, trees, and free quasi-symmetric functions, Journal of Pure and Applied Algebra, vol.209, issue.2, pp.439-459, 2007.
DOI : 10.1016/j.jpaa.2006.06.005
URL : https://hal.archives-ouvertes.fr/hal-00004902

L. Foissy and C. Malvenuto, The Hopf algebra of finite topologies and T-partitions, Journal of Algebra, vol.438, pp.130-169, 2015.
DOI : 10.1016/j.jalgebra.2015.04.024
URL : https://hal.archives-ouvertes.fr/hal-01017229

L. Foissy and F. Patras, NATURAL ENDOMORPHISMS OF SHUFFLE ALGEBRAS, International Journal of Algebra and Computation, vol.23, issue.04, pp.989-1009, 2013.
DOI : 10.1142/S0218196713400183
URL : https://hal.archives-ouvertes.fr/hal-01289294

L. Foissy, F. Patras, and J. Thibon, D??formations des alg??bres de battages et quasi-battages, Annales de l???institut Fourier, vol.66, issue.1
DOI : 10.5802/aif.3010

I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh et al., Noncommutative Symmetrical Functions, Advances in Mathematics, vol.112, issue.2, pp.218-348, 1994.
DOI : 10.1006/aima.1995.1032

E. Getzler and J. D. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, arXiv preprint hep-th, p.9403055, 1994.

F. Hivert, J. Novelli, and J. Thibon, The algebra of binary search trees, Theoretical Computer Science, vol.339, issue.1, pp.129-165, 2005.
DOI : 10.1016/j.tcs.2005.01.012
URL : https://hal.archives-ouvertes.fr/hal-00622706

F. Hivert, J. Novelli, and J. Thibon, Trees, functional equations, and combinatorial Hopf algebras, Trees, functional equations, and combinatorial hopf algebras, pp.1682-1695, 2008.
DOI : 10.1016/j.ejc.2007.09.005
URL : https://hal.archives-ouvertes.fr/hal-00484666

M. E. Hoffman, Quasi-shuffle products, J. Algebraic Combin, vol.11, issue.1, pp.49-68, 2000.

M. Livernet and F. Patras, Lie theory for Hopf operads, Journal of Algebra, vol.319, issue.12, pp.4899-4920, 2008.
DOI : 10.1016/j.jalgebra.2008.03.016
URL : https://hal.archives-ouvertes.fr/hal-00083757

J. Loday, On the algebra of quasi-shuffles, manuscripta mathematica 123, pp.79-93, 2007.

J. Loday and M. Ronco, Hopf Algebra of the Planar Binary Trees, Advances in Mathematics, vol.139, issue.2, pp.293-309, 1998.
DOI : 10.1006/aima.1998.1759

C. Malvenuto, . Ch, and . Reutenauer, Duality between Quasi-Symmetrical Functions and the Solomon Descent Algebra, Journal of Algebra, vol.177, issue.3, pp.967-982, 1995.
DOI : 10.1006/jabr.1995.1336

J. Novelli, F. Patras, and J. Thibon, Natural endomorphisms of quasi-shuffle Hopf algebras, Bulletin de la Société mathématique de France, vol.141, issue.1, pp.107-130, 2013.
DOI : 10.24033/bsmf.2644
URL : https://hal.archives-ouvertes.fr/hal-00823090

J. Novelli and J. Thibon, Polynomial realizations of some trialgebras, Proceedings of Formal Power Series and Algebraic Combinatorics, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00622743

F. Patras, La décomposition en poids des algebres de hopf, Annales de l'institut Fourier 43 29. , L'algèbre des descentes d'une bigèbre graduée, Journal of Algebra, vol.170, issue.4 2, pp.1067-1087, 1993.

F. Patras, . Ch, and . Reutenauer, On descent algebras and twisted bialgebras, Mosc. Math. J, vol.4, issue.1, pp.199-216, 2004.

F. Patras and M. Schocker, Twisted descent algebras and the Solomon???Tits algebra, Advances in Mathematics, vol.199, issue.1, pp.151-184, 2006.
DOI : 10.1016/j.aim.2005.01.010
URL : https://hal.archives-ouvertes.fr/hal-00128596

F. Patras and M. , Trees, set compositions and the twisted descent algebra, Journal of Algebraic Combinatorics, vol.86, issue.3, pp.3-23, 2008.
DOI : 10.1007/s10801-006-0028-1

. Ch and . Reutenauer, Free lie algebras, 1993.

G. Rota, . Baxter, . Bull, and . Amer, Fluctuation theory and baxter algebras, Math. Soc, vol.325, issue.75, pp.325-330, 1969.

M. P. Schützenberger, Sur une propriété combinatoire des algèbres de lie libres pouvantêtrevantêtre utilisée dans unprobì eme de mathématiques appliquées, Séminaire Dubreil? Jacotin Pisot, vol.59, 1958.

B. P. Laboratoire-de-mathématiques, 1039, 51687 Reims Cedex 2, France E-mail address: loic.foissy@univ-reims.fr UMR 6621 CNRS