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Natural endomorphisms of shuffle algebras
 International Journal of Algebra and Computation
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Ecalle’s arborification–coarborification transforms and Connes–Kreimer Hopf algebra
, 2012
"... We give a natural and complete description of Ecalle’s mould–comould formalism within a Hopf–algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes–Kreime ..."
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We give a natural and complete description of Ecalle’s mould–comould formalism within a Hopf–algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes–Kreimer Hopf algebra. We give a straightforward characterization of the fundamental process of homogeneous coarborification, using the explicit duality between decorated Connes–Kreimer and Grossman– Larson algebras. Finally, we introduce a new Hopf algebra that systematically underlies the calculations for the normalization of local dynamical systems.
Flows on rooted trees and the Narayana idempotents
, 2012
"... Several generating series for flows on rooted trees are introduced, as elements in the group of series associated with the PreLie operad. By combinatorial arguments, one proves identities that characterise these series. One then gives a complete description of the image of these series in the grou ..."
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Several generating series for flows on rooted trees are introduced, as elements in the group of series associated with the PreLie operad. By combinatorial arguments, one proves identities that characterise these series. One then gives a complete description of the image of these series in the group of series associated with the Dendriform operad. This allows to recover the Lie idempotents in the descent algebras recently introduced by Menous, Novelli and Thibon. Moreover, one defines new Lie idempotents and conjecture the existence of some others.
A setoperad of formal . . .
, 2013
"... We introduce an operad of formal fractions, abstracted from th eMould operads and containing both the Dendriform and the Tridendriform operads. We consider the smallest setoperad contained in this operad and containing four specific elements of arity two, corresponding to the generators and the ass ..."
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We introduce an operad of formal fractions, abstracted from th eMould operads and containing both the Dendriform and the Tridendriform operads. We consider the smallest setoperad contained in this operad and containing four specific elements of arity two, corresponding to the generators and the associative elements of the Dendriform and Tridendriform operads. We obtain a presentation of this operad (by binary generators and quadratic relations) and an explicit combinatorial description using a new kind of bicolored trees. Similar results are also
NATURAL ENDOMORPHISMS OF SHUFFLE ALGEBRAS
, 2012
"... Shuffles have a long history, starting with the probabilistic study of card shufflings in the first part of the 20th century by Borel, Hadamard, Poincaré and others. Their theory was revived in the 50’s, for various reasons. In topology, the combinatorics of (non commutative) shuffle products was th ..."
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Shuffles have a long history, starting with the probabilistic study of card shufflings in the first part of the 20th century by Borel, Hadamard, Poincaré and others. Their theory was revived in the 50’s, for various reasons. In topology, the combinatorics of (non commutative) shuffle products was the