On the enumeration and generation of generalized Dyck words

Duchon, Philippe

doi:10.1016/s0012-365x(00)00150-3

Cited by 77 publications

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“…It is now understood that the kernel method provides a systematic solution to all such problems, which are, in essence, one-dimensional [14,2]. Other generic approaches to half-space problems are provided in [20,25].…”

Section: Introductionmentioning

confidence: 99%

Walks with small steps in the quarter plane

Bousquet-Mélou¹,

Mishna²

2010

Algorithmic Probability and Combinatorics

159

569

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Abstract. Let S ⊂ {−1, 0, 1} 2 \ {(0, 0)}. We address the enumeration of plane lattice walks with steps in S, that start from (0, 0) and always remain in the first quadrant {(i, j) : i ≥ 0, j ≥ 0}. A priori, there are 2 8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a half-plane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study.To each of them, we associate a group G of birational transformations. We show that this group is finite (of order at most 8) in 23 cases, and infinite in the 56 other cases.We present a unified way of solving 22 of the 23 models associated with a finite group. For all of them, the generating function is found to be D-finite. The 23rd model, known as Gessel's walks, has recently been proved by Bostan et al. to have an algebraic (and hence D-finite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a non-D-finite generating function.Our approach allows us to recover and refine some known results, and also to obtain new results. For instance, we prove that walks with N, E, W, S, SW and NE steps have an algebraic generating function.

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Section: Introductionmentioning

confidence: 99%

Walks with small steps in the quarter plane

Bousquet-Mélou¹,

Mishna²

2010

Algorithmic Probability and Combinatorics

159

569

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“…Thus the whole machinery we developed for M/M r /1 works after a proper reparameterization again. Hence all that has to be done is to exchange m ↔ k and λ ↔ µ in formula (11). We omit the details here, for more information see and Böhm (1993) where these models are dealt with in discrete time and the corresponding continuous time formulas are derived by a limiting procedure.…”

Section: Champernowne's Constructionmentioning

confidence: 99%

Lattice path counting and the theory of queues

Böhm

2010

Journal of Statistical Planning and Inference

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“…In the case at hand, the slope is 2m+1 2 . In [9], Duchon provided an algebraic grammar for generalized Dyck languages (as introduced by Labelle and Yeh [10]) and proved that words in such a language can be obtained uniquely by inserting words of the language into factor-free words of the same language.…”

Section: Introductionmentioning

confidence: 99%

“…As shown in [9,Section 5], the algebraic grammars for D A,h and D A,h can be described by a system of derivation rules in terms of certain auxiliary languages with restrictions on their total and partial valuations. As we will see in Section 2, for Dyck words with slope 2m+1 2 , the aforementioned derivation rules may be reduced to a single core language that we denote by U , or U 2m+1 2 if we wish to emphasize the slope.…”

Section: Introductionmentioning

confidence: 99%

On factor-free Dyck words with half-integer slope

Birmajer

Gil²,

Weiner³

2018

Advances in Applied Mathematics

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We study a class of rational Dyck paths with slope 2m+1 2 corresponding to factor-free Dyck words, as introduced by P. Duchon. We show that, for the slopes considered in this paper, the language of factor-free Dyck words is generated by an auxiliary language that we examine from the algebraic and combinatorial points of view. We provide a lattice path description of this language, and give an explicit enumeration formula in terms of partial Bell polynomials. As a corollary, we obtain new formulas for the number of associated factor-free generalized Dyck words.

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On the enumeration and generation of generalized Dyck words

Cited by 77 publications

References 0 publications

Walks with small steps in the quarter plane

Walks with small steps in the quarter plane

Lattice path counting and the theory of queues

On factor-free Dyck words with half-integer slope

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