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Shuffle of parenthesis systems and baxter permutations

Robert Cori, Serge Dulucq, Gérard Viennot
1986 Journal of combinatorial theory. Series A  
The only thing we have really to verify is that adding a "jj" at the end 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  ... 
doi:10.1016/0097-3165(86)90018-x fatcat:6m4rl2wcmjd5riwkxqzjraeeru

Stack words, standard tableaux and Baxter permutations

S. Dulucq, O. Guibert
1996 Discrete Mathematics  
In the second part, we recall the notion 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  ... 
doi:10.1016/s0012-365x(96)83009-3 fatcat:zy6yk43wejcfniq3qnmoqycfqq

Involutions on Baxter Objects [article]

Kevin Dilks
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  ... 
arXiv:1402.2961v1 fatcat:jxp4tzut2bfivpkz5x25im5i5u

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

S. Dulucq, O. Guibert
1998 Discrete Mathematics  
Chung et al. (1978) have proved that the number 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,~}.  ... 
doi:10.1016/s0012-365x(97)00112-x fatcat:z2w3gmm2lfdq7pwkvculnd5weu

Bijective counting of tree-rooted maps and shuffles of parenthesis systems [article]

Olivier Bernardi
2006 arXiv   pre-print
We prove that our bijection is isomorphic to a former recursive construction on 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.  ... 
arXiv:math/0601684v1 fatcat:6inhiu3jwvbaznis76pf2qljtm

Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems

Olivier Bernardi
2007 Electronic Journal of Combinatorics  
Then, we prove that our bijection is isomorphic to a former recursive construction on 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.  ... 
doi:10.37236/928 fatcat:i4kk2ys4rbccpkdtjdvy2lxevq

Lie theory for quasi-shuffle bialgebras [article]

Loïc Foissy
2016 arXiv   pre-print
to the beginings 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.  ... 
arXiv:1605.02444v1 fatcat:rozsx2rpnbdrxdzbvxpa7asaqy

Bijections for Baxter families and related objects

Stefan Felsner, Éric Fusy, Marc Noy, David Orden
2011 Journal of combinatorial theory. Series A  
n, 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.  ... 
doi:10.1016/j.jcta.2010.03.017 fatcat:l4kpf5aggnb5beocsyj77qsjmu

Free braided nonassociative Hopf algebras and Sabinin τ-algebras [article]

Ualbai Umirbaev, Vladislav Kharchenko
2020 arXiv   pre-print
We introduce a braided version 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).  ... 
arXiv:2001.00304v1 fatcat:pumntyh63ja7jgybofksdoxnvi

Interview with Xavier Viennot

ECA, University of Haifa
2021 Enumerative Combinatorics and Applications  
We did not know at the same time, Guttmann 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.  ... 
doi:10.54550/eca2022v2s1i4 fatcat:n2kaan37zrcc3l6xf25tsg477u

The Epstein–Glaser approach to perturbative quantum field theory: graphs and Hopf algebras

Alexander Lange
2005 Journal of Mathematical Physics  
The paper aims at investigating perturbative quantum field theory (pQFT) in the approach 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  ... 
doi:10.1063/1.1893215 fatcat:pgdwky3tybbfbfuuavh3fyx5lu

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]

David Lopez-Paz, Marc'Aurelio Ranzato
2022 arXiv   pre-print
Our experiments on variants 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.  ... 
arXiv:1706.08840v6 fatcat:mymiorqh55gynnml3q5cafc5qe
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