Pattern distributions in Dyck paths with a first return decomposition constrained by height

Baril, Jean-Luc; Genestier, Richard; Kirgizov, Sergey

doi:10.1016/j.disc.2020.111995

Cited by 5 publications

(5 citation statements)

References 19 publications

(16 reference statements)

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“…. , C i+1 )) = P , where P is the path described as This function is a bijection, where the inverse function is given by decomposing a path by using the following algorithm: Algorithm 1 Inverse function φ (or reverse) (1) If there are (−1)-valleys, go to step (2). If there are no consecutive valleys with difference equal to −1, then the path is increasing and can be decomposed by using only pyramids ∆ and λ in the following way:…”

Section: Number Of D-dyck Paths and Peaks Statisticmentioning

confidence: 99%

“…. , n − 1} there is exactly one path in Q n of the form (XY ) i (XY ) n−i , where its area is equal to i 2 + (n − i) 2 . So, the total area of these types of paths is…”

Section: The Area Of the (−1)-dyck Pathsmentioning

confidence: 99%

“…, ν k ) formed by all y-coordinates (listed from left to right) of all valley vertices of P . For further recent work about different combinatorial aspects of Dyck paths, see for instance [2,3,5,10,15,20].…”

Section: Introductionmentioning

confidence: 99%

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Restricted Dyck Paths on Valleys Sequence

Flórez¹,

Mansour²,

Ramı́rez³

et al. 2021

Preprint

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In this paper we study a subfamily of a classic lattice path, the Dyck paths, called restricted d-Dyck paths, in short d-Dyck. A valley of a Dyck path P is a local minimum of P ; if the difference between the heights of two consecutive valleys (from left to right) is at least d, we say that P is a restricted d-Dyck path. The area of a Dyck path is the sum of the absolute values of y-components of all points in the path. We find the number of peaks and the area of all paths of a given length in the set of d-Dyck paths. We give a bivariate generating function to count the number of the d-Dyck paths with respect to the the semi-length and number of peaks. After that, we analyze in detail the case d = −1. Among other things, we give both, the generating function and a recursive relation for the total area.

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Section: Number Of D-dyck Paths and Peaks Statisticmentioning

confidence: 99%

“…. , n − 1} there is exactly one path in Q n of the form (XY ) i (XY ) n−i , where its area is equal to i 2 + (n − i) 2 . So, the total area of these types of paths is…”

Section: The Area Of the (−1)-dyck Pathsmentioning

confidence: 99%

See 1 more Smart Citation

Restricted Dyck Paths on Valleys Sequence

Flórez¹,

Mansour²,

Ramı́rez³

et al. 2021

Preprint

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“…For n ě 1, we denote by n the constant statistic returning the value n. The popularity of a pattern p in S is the total number of occurrences of p over all objects of S, that is ppSq " ř aPS ppaq ( [6,9,10]). Let S 1 be a set of combinatorial objects, we say that two statistics, s on S and t on S 1 , have the same distribution if there exists a bijection f : S Ñ S 1 satisfying spaq " tpf paqq for any a P S. In this case, with a slight abuse of the notation already used in [5], we write f psq " t or s " t whenever f is the identity.…”

Section: Introduction and Notationsmentioning

confidence: 99%

Enumeration of Dyck paths with air pockets

Baril¹,

Kirgizov²,

Maréchal³

et al. 2022

Preprint

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We introduce and study the new combinatorial class of Dyck paths with air pockets. We exhibit a bijection with the peakless Motzkin paths which transports several pattern statistics and give bivariate generating functions for the distribution of patterns as peaks, returns and pyramids. Then, we deduce the popularities of these patterns and point out a link between the popularity of pyramids and a special kind of closed smooth self-overlapping curves, a subset of Fibonacci meanders. A similar study is conducted for the subclass of non-decreasing Dyck paths with air pockets.

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“…We say that two statistics s ∈ T A and t ∈ T B have the same distribution, or are equidistributed, if there exists a bijection f : A → B such that s(a) = t(f (a)) for any a ∈ A. In this case, with a slight abuse of the notation already used in [4], we write shortly f (s) = t or s = t whenever f is the identity.…”

Section: Introductionmentioning

confidence: 99%

Pattern statistics in faro words and permutations

Baril¹,

Burstein²,

Kirgizov³

2020

Preprint

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We study the distribution and the popularity of some patterns in words obtained by interlacing the letters of the two nondecreasing k-ary words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths with a given number of peaks. We show how the bijection maps statistics of consecutive patterns into linear combinations of other pattern statistics on paths. We deduce enumerative results by providing multivariate generating functions for the distribution and the popularity of patterns of length at most three. Finally, we consider some interesting subclasses of faro words that are permutations, involutions, derangements, or subexcedent words.

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Pattern distributions in Dyck paths with a first return decomposition constrained by height

Cited by 5 publications

References 19 publications

Restricted Dyck Paths on Valleys Sequence

Restricted Dyck Paths on Valleys Sequence

Enumeration of Dyck paths with air pockets

Pattern statistics in faro words and permutations

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