Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d'euler et nombres de Genocchi

Francon, J; Viennot, Gérard

doi:10.1016/0012-365x(79)90182-1

Cited by 99 publications

(101 citation statements)

References 9 publications

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“…We will show in Section 7 that the two kinds of paths are in bijection with signed permutations, using variants of classical bijections of Françon and Viennot [15], Foata and Zeilberger [14]. We obtain two other interpretations of B n (y, t, q) where y follows a descent statistic and q a pattern statistic.…”

Section: Definitions and Resultsmentioning

confidence: 99%

“…In this reference, the first author obtains refinements of two bijections originally given by Françon and Viennot [15], Foata and Zeilberger [14]. In the case of signed permutations, we will see that each of these two bijections has two variants, corresponding to the two kinds of paths obtained in the previous section.…”

Section: Interpretation Of B N (Y T Q) Via Weighted Motzkin Pathsmentioning

confidence: 99%

“…Then, we make use of a method called Matrix Ansatz, as in previous work on (unsigned) permutations and originally related with the PASEP. This has several consequences, in particular we relate our q-Eulerian numbers with combinatorics of weighted Motzkin paths (in the style of Flajolet [12] and Françon and Viennot [15]) and give several bijections with signed permutations, that give other combinatorial interpretations of our q-Eulerian numbers.…”

Section: Introductionmentioning

confidence: 99%

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Crossings of signed permutations and $q$-Eulerian numbers of type $B$

Corteel

Josuat-Vergès

Kim

2013

Journal of Combinatorics

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Abstract. In this paper we want to study combinatorics of the type B permutations and in particular the join statistics crossings, excedances and the number of negative entries. We generalize most of the results known for type A (i.e. zero negative entries) and use a mix of enumerative, algebraic and bijective techniques. This work has been motivated by permutation tableaux of type B introduced by Lam and Williams, and natural statistics that can be read on these tableaux. We mostly use (pignose) diagrams and labelled Motzkin paths for the combinatorial interpretations of our results.

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Section: Definitions and Resultsmentioning

confidence: 99%

Section: Interpretation Of B N (Y T Q) Via Weighted Motzkin Pathsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Crossings of signed permutations and $q$-Eulerian numbers of type $B$

Corteel

Josuat-Vergès

Kim

2013

Journal of Combinatorics

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“…-Motzkin numbers: this famous sequence of numbers arises in the solution of the cases S = {x, y, xȳ} and S = {x,x, y,ȳ, xȳ,xy} (Propositions 9 and 10). The first problem is equivalent to the enumeration of involutions with no decreasing subsequence of length 4, and the occurrence of Motzkin numbers follows from restricting a bijection of Françon and Viennot [24]. The solution to the second problem is, to our knowledge, new, and deserves a more combinatorial solution.…”

Section: Explain Closed Form Expressionsmentioning

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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“…Numérotons les sommets internes d'un arbre binaire b de taille n par les entiers 1,2, .. .,n en ordre préfixe (d'autres ordres peuvent convenir aussi, voir [6 ] et [7 ]). Associons à b le chemin c = n(b) de la façon suivante : pour Ï=1,2, .. .,n-1 posons ) = c(f)+l si les deux fils du sommet i sont internes ) -c(i) -1 si les deux fils du sommet i sont externes (i est feuille) ) = c(f) sinon et ce palier est coloré par + (resp.…”

Section: Une Relation De Récurrence Pour Les Nombres De Strahlerunclassified

Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d'euler et nombres de Genocchi

Cited by 99 publications

References 9 publications

Crossings of signed permutations and $q$-Eulerian numbers of type $B$

Crossings of signed permutations and $q$-Eulerian numbers of type $B$

Walks with small steps in the quarter plane

Sur le nombre de registres nécessaires à l'évaluation d'une expression arithmétique

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