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The Hopf algebra of rooted trees, free Lie algebras
 Email address: Philippe.Chartier@inria.fr Section de Mathématiques, Université de Genève, 24 rue du Lièvre, 1211 Genève 4, Switzerland Email address: Ernst.Hairer@unige.ch Section de Mathématiques, École Polytechnique Fédérale de Lausanne, SB
, 2006
"... We present an approach that allows performing computations related to the BakerCampbellHaussdorff (BCH) formula and its generalizations in an arbitrary Hall basis, using labeled rooted trees. In particular, we provide explicit formulas (given in terms of the structure of certain labeled rooted tre ..."
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Cited by 43 (9 self)
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We present an approach that allows performing computations related to the BakerCampbellHaussdorff (BCH) formula and its generalizations in an arbitrary Hall basis, using labeled rooted trees. In particular, we provide explicit formulas (given in terms of the structure of certain labeled rooted trees) of the continuous BCH formula. We develop a rewriting algorithm (based on labeled rooted trees) in the dual PoincaréBirkhoffWitt (PBW) basis associated to an arbitrary Hall set, that allows handling Lie series, exponentials of Lie series, and related series written in the PBW basis. At the end of the paper we show that our approach is actually based on an explicit description of an epimorphism ν of Hopf algebras from the commutative Hopf algebra of labeled rooted trees to the shuffle Hopf algebra and its kernel ker ν. 1 Introduction, general setting, and examples Consider a ddimensional system of nonautonomous ODEs of the form d
Combinatorics of rooted trees and Hopf algebras
, 2002
"... We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of nonroot vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the op ..."
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Cited by 40 (5 self)
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We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of nonroot vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities. Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators to Kreimer’s Hopf algebra and relate them via the inner product. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the GrossmanLarson Hopf algebra with the graded dual of Kreimer’s Hopf algebra, correcting an earlier result of Panaite. 1
QED Hopf algebras on planar binary trees
 Preprint 2001/15 of Institut Girard Desargues, arXiv:math.QA/0112043
"... In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the renormalization procedure. Both structures are related to some semidirect coproduct of Hopf algebras. 1 ..."
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Cited by 32 (7 self)
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In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the renormalization procedure. Both structures are related to some semidirect coproduct of Hopf algebras. 1
Profiles of random trees: Limit theorems for random recursive trees and binary search trees
, 2005
"... We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio ˛ of the level and the logarithm of tree size lies in Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 1 (with only con ..."
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Cited by 26 (11 self)
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We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio ˛ of the level and the logarithm of tree size lies in Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 1 (with only convergence of finite moments when ˛ 2.1; e/). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for ˛ D 0 and a “quicksort type ” limit law for ˛ D 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.
Structure of the LodayRonco Hopf algebra of trees
 J. ALGEBRA
"... Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of noncommutative symmetric functions in the MalvenutoReutenauer Hopf algebra of permutations factors through their Hopf algebra ..."
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Cited by 21 (3 self)
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Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of noncommutative symmetric functions in the MalvenutoReutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra. We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopfalgebraic structure. We also obtain a transparent proof of its isomorphism with the noncommutative ConnesKreimer Hopf algebra of Foissy, and show that this algebra is related to noncommutative symmetric functions as the (commutative) ConnesKreimer Hopf algebra is related to symmetric functions.
Profiles of random trees: correlation and width of random recursive trees and binary search trees
 ADVANCES IN APPLIED PROBABILITY
, 2004
"... We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp signchanges when one level is fixed and the other one is varying. An asymptotic estimate for the expected width is also derived. ..."
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Cited by 20 (7 self)
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We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp signchanges when one level is fixed and the other one is varying. An asymptotic estimate for the expected width is also derived.
Relating the ConnesKreimer and GrossmanLarson Hopf algebras built on rooted trees
"... In [8], Dirk Kreimer discovered the striking fact that the process of renormalization in quantum field theory may be described, in a conceptual manner, by means of certain Hopf algebras (which depend on the chosen renormalization scheme). A toy model was studied in detail by Alain Connes and Dirk Kr ..."
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Cited by 19 (0 self)
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In [8], Dirk Kreimer discovered the striking fact that the process of renormalization in quantum field theory may be described, in a conceptual manner, by means of certain Hopf algebras (which depend on the chosen renormalization scheme). A toy model was studied in detail by Alain Connes and Dirk Kreimer in [3]; the Hopf
Profiles of random trees: planeoriented recursive trees
, 2005
"... We derive several limit results for the profile of random planeoriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width and the correlation coefficients of two level sizes. Most of ..."
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Cited by 17 (5 self)
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We derive several limit results for the profile of random planeoriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width and the correlation coefficients of two level sizes. Most of our proofs are based on a method of moments. We also discover an unexpected connection between the profile of planeoriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size).
COMMUTATIVE COMBINATORIAL HOPF ALGEBRAS
, 2006
"... Abstract. We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its noncommutative du ..."
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Cited by 15 (5 self)
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Abstract. We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its noncommutative dual is realized in three different ways, in particular as the GrossmanLarson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce oneparameter families interpolating between different structures constructed on the same combinatorial objects. Contents
Cocommutative Hopf algebras of permutations and trees
 JOURNAL ALGEBRAIC COMBINATORICS
, 2004
"... Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980’s ..."
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Cited by 14 (4 self)
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Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980’s by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heapordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.