Planar rooted trees and non-associative exponential series

Combinatorics International audience A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.

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Prismatic Sets Associated with Planar Rooted Trees

Journal of Mathematics ◽

10.1155/2019/1543201 ◽

2019 ◽

Vol 2019 ◽

pp. 1-6

Author(s):

S. Kaan Gürbüzer ◽

Bedia Akyar

Keyword(s):

Rooted Trees ◽

Prismatic Structure ◽

Algebraic Operations ◽

Planar Rooted Trees

We construct the almost strong prismatic structure on the set of planar rooted trees and the bicomplex of planar rooted trees. Furthermore, we study the prismatic properties of Loday’s algebraic operations on the set of planar rooted trees.

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Constructing combinatorial operads from monoids

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3034 ◽

2012 ◽

Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽

Author(s):

Samuele Giraudo

Keyword(s):

Rooted Trees ◽

Parking Functions ◽

Combinatorial Objects ◽

Dyck Paths ◽

International Audience ◽

Motzkin Paths ◽

Planar Rooted Trees

International audience We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

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Hopf algebras on planar trees and permutations

Journal of Algebra and Its Applications ◽

10.1142/s0219498822502243 ◽

2021 ◽

Author(s):

Diego Arcis ◽

Sebastián Márquez

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Rooted Trees ◽

Labeled Trees ◽

Different Types ◽

Structure Lead ◽

Planar Rooted Trees

We endow the space of rooted planar trees with the structure of a Hopf algebra. We prove that variations of such a structure lead to Hopf algebras on the spaces of labeled trees, [Formula: see text]-trees, increasing planar trees and sorted trees. These structures are used to construct Hopf algebras on different types of permutations. In particular, we obtain new characterizations of the Hopf algebras of Malvenuto–Reutenauer and Loday–Ronco via planar rooted trees.

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On the Lie enveloping algebra of a pre-Lie algebra

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is008001011jkt037 ◽

2008 ◽

Vol 2 (1) ◽

pp. 147-167 ◽

Cited By ~ 23

Author(s):

J.-M. Oudom ◽

D. Guin

Keyword(s):

Hopf Algebra ◽

Lie Algebra ◽

Lie Algebras ◽

Rooted Trees ◽

Enveloping Algebra ◽

Associative Product ◽

Planar Rooted Trees ◽

Symmetric Brace Algebras

AbstractWe construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

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Typed Angularly Decorated Planar Rooted Trees and Ω-Rota-Baxter Algebras

Mathematics ◽

10.3390/math10020190 ◽

2022 ◽

Vol 10 (2) ◽

pp. 190

Author(s):

Yi Zhang ◽

Xiaosong Peng ◽

Yuanyuan Zhang

Keyword(s):

Rooted Trees ◽

Integral Analysis ◽

Dendriform Algebra ◽

Dendriform Algebras ◽

Definition Of ◽

The Relationship ◽

Planar Rooted Trees

As a generalization of Rota–Baxter algebras, the concept of an Ω-Rota–Baxter could also be regarded as an algebraic abstraction of the integral analysis. In this paper, we introduce the concept of an Ω-dendriform algebra and show the relationship between Ω-Rota–Baxter algebras and Ω-dendriform algebras. Then, we provide a multiplication recursion definition of typed, angularly decorated rooted trees. Finally, we construct the free Ω-Rota–Baxter algebra by typed, angularly decorated rooted trees.

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Multisource Broadcasting on de Bruijn and Kautz Digraphs Using Isomorphic Factorizations into Cycle-Rooted Trees

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e92.a.1757 ◽

2009 ◽

Vol E92-A (8) ◽

pp. 1757-1763

Author(s):

Takahiro TSUNO ◽

Yukio SHIBATA

Keyword(s):

Rooted Trees ◽

Kautz Digraphs ◽

De Bruijn

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On The Tails of the Exponential Series

Canadian Mathematical Bulletin ◽

10.4153/cmb-1994-041-5 ◽

1994 ◽

Vol 37 (2) ◽

pp. 278-286 ◽

Cited By ~ 5

Author(s):

C. Yalçin Yildirim

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Partial Sums ◽

Maclaurin Series ◽

Exponential Series ◽

Asymptotic Estimation ◽

The Riemann Zeta Function ◽

Riemann Zeta

AbstractA relation between the zeros of the partial sums and the zeros of the corresponding tails of the Maclaurin series for ez is established. This allows an asymptotic estimation of a quantity which came up in the theory of the Riemann zeta-function. Some new properties of the tails of ez are also provided.

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