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Shuffle of parenthesis systems and baxter permutations

1986
*
Journal of combinatorial theory. Series A
*

The only thing we have really to verify is that adding a "jj" at the end

doi:10.1016/0097-3165(86)90018-x
fatcat:6m4rl2wcmjd5riwkxqzjraeeru
*of*a*parenthesis**system**and*a "y" at the begin we obtain also a*parenthesis**system*. ... The mapping a -+ (C(a), D(a)) is a bijection: Example 6. 8 .FIG. 6 . 86 Let c1= abiiba@ibiib be a word*of*length 2n = 10*of*Pab (*shuffle**of*the two*parenthesis*words aiiaii*and*bb666& The*Baxter*tree ...##
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Stack words, standard tableaux and Baxter permutations

1996
*
Discrete Mathematics
*

In the second part, we recall the notion

doi:10.1016/s0012-365x(96)83009-3
fatcat:zy6yk43wejcfniq3qnmoqycfqq
*of**shuffle**of*two*parenthesis**systems**and*the correspondence exhibited by Cori et al. [3] between these words, couples*of*complete binary trees*and*alt~ting*Baxter*... Cori et al. [3] have established a bijection, that we denote T, between the language Sht@e2,,*shuffling*two*parenthesis*languages, the set B^axferzn*of*alternating*Baxter**permutations*,*and*the set*of*all ...##
###
Involutions on Baxter Objects
[article]

2014
*
arXiv
*
pre-print

*Baxter*numbers are known to count several families

*of*combinatorial objects, all

*of*which come equipped with natural involutions. ... In this paper, we add a combinatorial family to the list,

*and*show that the known bijections between these objects respect these involutions. ... Additionally, if we think

*of*1's as being z's (corresponding to left parentheses)

*and*2's as being z's (corresponding to right parentheses), then P z,z , the language

*of*well-formed

*parenthesis*

*systems*...

##
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Page 6093 of Mathematical Reviews Vol. , Issue 87k
[page]

1987
*
Mathematical Reviews
*

Cori, Robert (F-BORD); 87k:05010 Dulucgq, Serge (F-BORD); Viennot, Gérard (F-BORD)

*Shuffle**of**parenthesis**systems**and**Baxter**permutations*. J. Combin. Theory Ser. A 43 (1986), no. 1, 1-22. ... In the article under review, the class*of*alternating*Baxter**permutations*is studied, i.e.*permutations*which satisfy (B1)*and*(B2) above*and*which are at the same time alternating in the classical sense ...##
###
Baxter permutations

1998
*
Discrete Mathematics
*

Chung et al. (1978) have proved that the number

doi:10.1016/s0012-365x(97)00112-x
fatcat:z2w3gmm2lfdq7pwkvculnd5weu
*of**Baxter**permutations*on [n] is ("r+1) u+I u+I \r+l } \r+2] ... [3] , by solving a question asked by Mullin [11] about enumeration*of*some planar maps, have established a one-to-one correspondence between the language*shuffle**of*two well-formed*parenthesis**systems*... Let Tree. be the set*of*binary trees having n vertices*and*Pz,~ the language*of*well-formed*parenthesis**systems*(or Dyck words) defined on the alphabet {z,~}. ...##
###
Bijective counting of tree-rooted maps and shuffles of parenthesis systems
[article]

2006
*
arXiv
*
pre-print

We prove that our bijection is isomorphic to a former recursive construction on

arXiv:math/0601684v1
fatcat:6inhiu3jwvbaznis76pf2qljtm
*shuffles**of**parenthesis**systems*due to Cori, Dulucq*and*Viennot. ... The number*of*tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree,*of*size n is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. ... Acknowledgments: I am deeply indebted to Mireille Bousquet-Mélou for struggling with several versions*of*this paper*and*coming out with very helpful suggestions. ...##
###
Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems

2007
*
Electronic Journal of Combinatorics
*

Then, we prove that our bijection is isomorphic to a former recursive construction on

doi:10.37236/928
fatcat:i4kk2ys4rbccpkdtjdvy2lxevq
*shuffles**of**parenthesis**systems*due to Cori, Dulucq*and*Viennot. ... The number*of*tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree,*of*size $n$ is ${\cal C}_{n} {\cal C}_{n+1}$ where ${\cal C}_{n}={1\over n+1}{2n \choose n}$ is the $n^{th ... Acknowledgments: I am deeply indebted to Mireille Bousquet-Mélou for struggling with several versions*of*this paper*and*coming out with very helpful suggestions. ...##
###
Lie theory for quasi-shuffle bialgebras
[article]

2016
*
arXiv
*
pre-print

to the beginings

arXiv:1605.02444v1
fatcat:rozsx2rpnbdrxdzbvxpa7asaqy
*of*the theory*of*Rota--*Baxter*algebras, but was developed systematically only recently, starting essentially with Hoffman's work, that was motivated by multizeta values (MZVs)*and*featured ... develop systematically a complete theory.This article builds on these various results*and*develops the analog theory, for quasi-*shuffle*algebras,*of*the theory*of*descent algebras*and*their relations to ... This article is, among others, a follow up*of*our joint works [26, 15] . We also thank the ICMAT Madrid for its hospitality. ...##
###
Bijections for Baxter families and related objects

2011
*
Journal of combinatorial theory. Series A
*

n,

doi:10.1016/j.jcta.2010.03.017
fatcat:l4kpf5aggnb5beocsyj77qsjmu
*and*Θ k, is the number*of**Baxter**permutations*with k descents*and*rises. ... The*Baxter*number B n can be written as B n = n k=0 Θ k,n−k−1 with These numbers have first appeared in the enumeration*of*so-called*Baxter**permutations*; B n is the number*of**Baxter**permutations**of*size ... Acknowledgments Mireille Bousquet-Mélou*and*Nicolas Bonichon are greatly thanked for fruitful discussions. We also thank an anonymous referee whose recommendations helped to improve the exposition. ...##
###
Free braided nonassociative Hopf algebras and Sabinin τ-algebras
[article]

2020
*
arXiv
*
pre-print

We introduce a braided version

arXiv:2001.00304v1
fatcat:pumntyh63ja7jgybofksdoxnvi
*of*Sabinin algebras*and*prove that the set*of*all primitive elements*of*a nonassociative τ-algebra is a Sabinin τ-algebra. ... In the case*of*involutive braidings, τ^2= id, we describe braided analogues*of*Shestakov-Umirbaev operations*and*prove that these operations are primitive operations. ... Let S n be the symmetric group on the set*of*symbols {1, 2, . . . , n}. A*permutation*π ∈ S n is called an r-*shuffle*if π(1) < π(2) < . . . < π(r), π(r + 1) < π(r + 2) < . . . < π(n). ...##
###
Interview with Xavier Viennot

2021
*
Enumerative Combinatorics and Applications
*

We did not know at the same time, Guttmann

doi:10.54550/eca2022v2s1i4
fatcat:n2kaan37zrcc3l6xf25tsg477u
*and*Enting were doing some experiments,*and*from the first values*of*the sequence, applying techniques*of*approximants, guess (*and*conjecture) the formula 124 ... This was the starting point*of*nice*and*fruitful cooperation between our combinatorial group in LaBRI*and*the department*of*statistical mechanics at Melbourne University. ... With this methodology, the differential*system**of*equations becomes a*system**of*non-commutative power series, where the product is replaced by the*shuffle*product. ...##
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The Epstein–Glaser approach to perturbative quantum field theory: graphs and Hopf algebras

2005
*
Journal of Mathematical Physics
*

The paper aims at investigating perturbative quantum field theory (pQFT) in the approach

doi:10.1063/1.1893215
fatcat:pgdwky3tybbfbfuuavh3fyx5lu
*of*Epstein*and*Glaser (EG)*and*, in particular, its formulation in the language*of*graphs*and*Hopf algebras (HAs) ... Various HAs are encountered, each one associated with a special combination*of*physical concepts such as normalization, localization, pseudo-unitarity, causality*and*an associated regularization,*and*renormalization ... I profited a lot from the discussions with him*and*with Kurush ...##
###
Page 1356 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

*Shuffle*

*of*

*parenthesis*

*systems*

*and*

*Baxter*

*permutations*. J. Combin. Theory Ser. A 43 (1986), no. 1, 1-22. (Volker Strehl) 87k:05010 05A15 (05A05, 05C05, 20B05) 34 (1987), no. 1-2, Viertl, R. ... (English summary) [Preservation

*of*minimality by random sampling] Analysis

*and*optimization

*of*

*systems*, Part 2 (Nice, 1984), 130-141, Lecture Notes in Control

*and*Inform. ...

##
###
Page 303 of Mathematical Reviews Vol. , Issue Index
[page]

*
Mathematical Reviews
*

(From the introduction) 87k:68075 § 68Q45 (05C15, 68R10, 90B10)
(with Cori, Robert; Viennot, Gérard)

*Shuffle**of**parenthesis**systems**and**Baxter**permutations*. J. Combin. Theory Ser. ... A parameter estimate associated with the adaptive control*of*stochastic*systems*. Analysis*and*optimization*of**systems*(Antibes, 1986), 508-514, Lecture Notes in Control*and*Inform. ...##
###
Gradient Episodic Memory for Continual Learning
[article]

2022
*
arXiv
*
pre-print

Our experiments on variants

arXiv:1706.08840v6
fatcat:mymiorqh55gynnml3q5cafc5qe
*of*the MNIST*and*CIFAR-100 datasets demonstrate the strong performance*of*GEM when compared to the state-*of*-the-art. ... To better understand this issue, we study the problem*of*continual learning, where the model observes, once*and*one by one, examples concerning a sequence*of*tasks. ... Olivier*and*A. Szlam for their insight. We are grateful to Martin Arjovsky for the QP interpretation*of*GEM. ...
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