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Equations with a dyck language solution

Topi Urponen
1976 Information and Control  
In cases where the solutions are unique these may be represented in terms of Dyck languages. We will also discuss equations with nonunique solutions. 1. PRELIMINARIES 1.1.  ...  In this paper we consider languages which are solutions of equations of the type X = XAiXBiX 4-"" + XA~XB~X + L a .  ...  In Sections 3 and 4 we consider equations with nonunique solutions. 1.2.  ... 
doi:10.1016/s0019-9958(76)90436-8 fatcat:5yxzrqg2obbcxikkxq7napte2u

Enumeration of polyominoes using Macsyma

M. Delest
1991 Theoretical Computer Science  
This paper shows the use of a symbolic language, Macsyma, to obtain new exact or asymptotic results il; combinatorics. The examples are taken among poiyominoes objects.  ...  Appendix 11 We give in the following a system of equations allowing the generation of the pigmented Dyck language defined in [9]. e denote by D the classical Dyck language.  ...  The letter R means that the woi-ds of the associated language to the non-terminal are prime Dyck words shuffled with letters "a". The non-terminal PP is associated to the pigmented Dyck language.  ... 
doi:10.1016/0304-3975(91)90152-r fatcat:ihw3prbh3zasng2ujpjybavozm

On the enumeration and generation of generalized Dyck words

Philippe Duchon
2000 Discrete Mathematics  
A random generation scheme is shown for generalized Dyck languages, which leads to some asymptotic results.  ...  We provide a new algebraic grammar for generalized Dyck languages as introduced by Labelle and Yeh (Discrete Math. 82 (1990) 1-6).  ...  Acknowledgements We would like to thank Mireille Bousquet-MÃ elou for pointing to us the right use of the Lagrange inversion formula, which lead to the enumeration formula for 3 2 -Dyck paths.  ... 
doi:10.1016/s0012-365x(00)00150-3 fatcat:7qa3fbqievbyffmwhz3tfdgymm

Page 565 of Mathematical Reviews Vol. 58, Issue 1 [page]

1979 Mathematical Reviews  
Representations of the Dyck languages using only language-theo- retic operations are given.  ...  Branislav Rovan (Bratislava) 58 #3747 Ruohonen, Keijo A note on language equations involving morphisms. Inform. Process. Lett. 7 (1978), no. 5, 209-212.  ... 

A note on the iteration of infinite matrices

Wolfgang Wechler
1988 RAIRO - Theoretical Informatics and Applications  
Assume that seR is a solution of the équation x -a + r.x for a,reR, that means s = a + r.s. Then r* exists and s = r*,a holds.  ...  Latter on we will see that 93* exists and its coefficient at the place (0,0) equals the Dyck language of order 1 with a and a as generators. • ITERATION OF INFINITE MATRICES A semiring R is called itérative  ...  ., a k , then the solution of the équation (*) equals the Dyck languages of order k {cf. [1] ).  ... 
doi:10.1051/ita/1988220201631 fatcat:5rd47rxqw5azncgh2gwfk4kykq

Zeta functions and topological entropy of the Markov-Dyck shifts [article]

Wolfgang Krieger, Kengo Matsumoto
2007 arXiv   pre-print
The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in Keller (G.  ...  For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topological entropy ae determined.  ...  Equation (4.23) is a multiple of equation (4.21) precisely if c = a + b and z = 1 1+a+b , and from this one sees, consulting (4.24), that both solutions of equation (4.21) are also solutions of equation  ... 
arXiv:0706.3262v1 fatcat:oyokjrefxfhzjo33rm5qhh6zhm

A characterization of poly-slender context-free languages

Lucian Ilie, Grzegorz Rozenberg, Arto Salomaa
2000 RAIRO - Theoretical Informatics and Applications  
We give a precise characterization of the kpoly-slender context-free languages. The well-known characterization of the k-poly-slender regular languages is an immediate consequence of ours.  ...  For a non-negative integer k, we say that a language L is k-poly-slender if the number of words of length n in L is of order O(n k ).  ...  Let us consider a context-free language L which is k-poly-slender. Then, by Theorem 6, L is a nite union of Dyck loops. If L contains no l-Dyck loop with l > k + 1, then we are done.  ... 
doi:10.1051/ita:2000100 fatcat:u5cp7v5dancjvn6hw2rsdhf6wa

Algebraic languages and polyominoes enumeration

Marie-Pierre Delest, Gérard Viennot
1984 Theoretical Computer Science  
The unique solution of the system contains the (noncommutative) generating function t = C H,E L w of the language L.  ...  By constructing a bijection between convex polyominoes and words of an algebraic language, atjd bq solving the corresponding algebraic qstem, we proie that the number of convex polyominoes with perimeter  ...  of equations, in term of the generating function c(t) of the Dyck language.  ... 
doi:10.1016/0304-3975(84)90116-6 fatcat:r7db4tihprfr5geedwhrww3l7q

Unary multiple equality sets: The languages of rational matrices

Franz-Josef Brandenburg
1981 Information and Control  
Finally crossing-free unary multiple equality sets are introduced, which vary from unary multiple equality sets in a similar way as the one-sided Dyck set varies from the two-sided Dyck set.  ...  These languages can equally well be defined by rational matrices. Unary multiple equality sets have extraordinary language theoretic properties.  ...  In a very natural way unary multiple equality sets can be associated with the sets of nonnegative integral solutions of systems of linear equations.  ... 
doi:10.1016/s0019-9958(81)90335-1 fatcat:xwgspui4avfvnpfnhk46wc2r7m

New Formulas for Dyck Paths in a Rectangle [article]

Jose Eduardo Blazek
2015 arXiv   pre-print
By using Ferrers diagrams associated to Dyck paths, we derive formulas for the enumeration of D_a,b with a and b non relatively prime, in terms of Catalan numbers.  ...  They are a natural generalization of the classical Dyck words enumerated by the Catalan numbers.  ...  It is well-known that the language of Dyck words on Σ = {0, 1} is the language generated by the algebraic grammar D → 0D1D + ε.  ... 
arXiv:1509.07832v1 fatcat:erinsx2cm5gybjllxtm4zw2q2m

Page 794 of Mathematical Reviews Vol. 49, Issue 2 [page]

1975 Mathematical Reviews  
It is shown that the parallel context-free languages are incomparable with the context-free languages (CFL) (ef. the example given above, and a direct proof is given that | the Dyck language is not PCL  ...  The author points out that a context-free language can be viewed as the minimal solution of a finite system of equations EH, of the form X,=f,(Xo, ---, X,, Yo, «++, Ym), where the X,, Y, range over subsets  ... 

Page 5455 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
A. (2-KIEV) Minimal elements of Church-Rosser systems with respect to formal languages.  ...  Summary: “The aim of this paper is to study the properties of languages generated by shuffle expressions with no ®@ operator nested.  ... 

Two combinatorial statistics on Dyck paths

Alain Denise, Rodica Simion
1995 Discrete Mathematics  
, Universit6 Bordeaux I, and with partial support from NSF grant DMS-9108749. 0012-365X/95/$09.50 C(C) 1995--Elsevier Science B.V.  ...  All rights reserved SSDI 0012-365X(93)E0147-V Definitions, notation, and preliminary results Dyck paths, pyramid weight, and exterior pairs A Dyck path is a path in the first quadrant, which begins at  ...  The reader will recognize the source of these equations: the second equation describes the Dyck paths of pyramid weight k > 0 via their unique factorization p=p'p" with p' being the maximal left primitive  ... 
doi:10.1016/0012-365x(93)e0147-v fatcat:2dntsivzg5gu5lsckove2ml6ka

Recursive Prime Factorizations: Dyck Words as Numbers [article]

Ralph L. Childress
2021 arXiv   pre-print
Dyck language represents a number.  ...  I propose a class of numeral systems where numbers are represented by Dyck words, with the systems arising from a generalization of prime factorization.  ...  Dyck language.  ... 
arXiv:2102.02777v2 fatcat:xs26cpa25banxlonu3hx6ovtuy

Attribute grammars are useful for combinatorics

M.P. Delest, J.M. Fedou
1992 Theoretical Computer Science  
The peaks (troughs) of a Dyck path correspond with the factors xx (xx) of the associated Dyck word. _I_ I -I_ _l _ l_ _I W(P)=S'(P) -_-_ -Fig. 5.  ...  Expanding the analytic solution of the previous equation shows that C, Definiton 3. 2 . 2 The q-analog of the generating function L(G) (denoted by 4L) is defined by qL = y& (w 4).  ... 
doi:10.1016/0304-3975(92)90380-x fatcat:eezr7g6pyraitoxz2uvceo5ita
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