Le treillis des chemins de Delannoy

Autebert, Jean-Michel; Latapy, Matthieu; Schwer, Sylviane

doi:10.1016/s0012-365x(02)00351-5

Cited by 5 publications

(8 citation statements)

References 6 publications

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“…Our goal is to begin a systematic study of the posets of paths arising in this way. A similar point of view has been undertaken in [1], where the authors study the posets arising from Delannoy paths; however, the present work deals with essentially different classes of paths. The only general problem we tackle here is to determine in which cases a poset of paths is a lattice, and we propose a possible solution to this problem.…”

Section: Introductionmentioning

confidence: 88%

Lattices of lattice paths

Ferrari

Pinzani

2005

Journal of Statistical Planning and Inference

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Section: Introductionmentioning

confidence: 88%

Lattices of lattice paths

Ferrari

Pinzani

2005

Journal of Statistical Planning and Inference

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“…Finally, there are some books [16,22,44,58,59,61,62,63] (Lucas, Frolow, and Catalan intensively corresponded with Delannoy for their books) or articles [2,12,14,5,49,50,51,54,65,69,70,71,74,75,78,83,87,88,89] which mention either Delannoy numbers or some of Delannoy's results/methods.…”

Section: 11mentioning

confidence: 99%

Why Delannoy numbers?

Banderier¹,

Schwer²

2005

Journal of Statistical Planning and Inference

114

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Abstract. This article is not a research paper, but a little note on the history of combinatorics: We present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers to the question raised in the title, as these works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the first general way to solve Ballot-like problems.

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“…If U (1) shares the transitivity property, U (1) = U * holds, and hence M (h) U * . Otherwise, the preceding lemma shows that M (h) U (2) , and iterating until U (i) = U * , in all cases, M (h) U * holds. U * being the matrix associated with f g, f g h is true.…”

mentioning

confidence: 96%

“…We proved formerly in [2] that L(p 1 , p 2 ) with the order relation −→ * is a distributive lattice.…”

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confidence: 99%

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On Generalized Delannoy Paths

Autebert¹,

Schwer²

2003

SIAM J. Discrete Math.

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A Delannoy path is a minimal path with diagonal steps in Z 2 between two arbitrary points. We extend this notion to the n dimensions space Z n and identify such paths with words on a special kind of alphabet: an S-alphabet. We show that the set of all the words corresponding to Delannoy paths going from one point to another is exactly one class in the congruence generated by a Thue system that we exhibit. This Thue system induces a partial order on this set that is isomorphic to the set of ordered partitions of a fixed multiset where the blocks are sets with a natural order relation. Our main result is that this poset is a lattice. Introduction. A Delannoy path[11] is given as a path that can be drawn on a rectangular grid, starting from the southwest corner, going to the northeast corner, using only three kinds of elementary steps: north, east, and northeast. Hence they are minimal paths with diagonal steps. We generalize the notion of a Delannoy path to the hyperspace Z n , considering a hyperparallelipedic grid as a set of elementary steps: a step in each direction and the combinations of several of them, the diagonal steps.We prove that, in a very natural way, an S-alphabet can be associated with the possible elementary steps in a Delannoy path in Z n , and consequently S-words with Delannoy paths themselves. These notions were introduced by Schwer [8], in a completely different context, for treating simultaneity problems.We then define a Thue system on the set of S-words that turns out to be noetherian and confluent. This Thue system induces both an ordering on S-words and a congruence. Our main goal is to prove that each equivalence class for this congruence is with this order relation a lattice (Theorem 5.5). (This lattice is a nondistributive lattice as soon as n > 2.) An equivalence class can be viewed as the set of all ordered partitions of a fixed multiset where the blocks are sets (not multisets). There is a transparent bijection between an equivalence class and an element of this set, and the order relation over partitions derived is a very natural one. In [9] are given some links between S-words and others mathematical objects.Moreover, we exhibit a characterization of the S-words of a class (and so of generalized Delannoy paths going from a point to another) with a family of matrices having its coefficients in {−1, 0, 1} (Theorem 4.2), and we prove that the order on S-words can be exactly transposed as the componentwise order on matrices induced by −1 < 0 < 1 (Theorem 4.6).

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Le treillis des chemins de Delannoy

Cited by 5 publications

References 6 publications

Lattices of lattice paths

Lattices of lattice paths

Why Delannoy numbers?

On Generalized Delannoy Paths

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