Nonleft peaks in Dyck paths: A combinatorial approach

Manes, Kostas; Sapounakis, A.; Tasoulas, I.; Tsikouras, P.

doi:10.1016/j.disc.2014.07.015

Cited by 6 publications

(4 citation statements)

References 8 publications

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“…See Figure 1 for detailed illustrations. In the literature, there are many papers dedicated to statistics of Dyck paths (words), see [1,2,3,4,5,6,7,8,9,10,11], [15,16] and the references therein. Recently, Flóres and Ramírez [8] find a formula for the total number, sp(n), of symmetric peaks over all Dyck paths of length 2(n + 1), as well as for the total number, ap(n), of asymmetric peaks over all Dyck paths of length 2(n + 3).…”

Section: Introductionmentioning

confidence: 99%

“…Manes, Sapounakis, Tasoulas and Tsikouras [10] study nonleft peaks in Dyck paths, which are peaks such that the u steps preceding them are greater than or equal to the d steps following them. They present a combinatorial construction of the set of Dyck paths of fixed length and number of nonleft peaks, and obtain various results on the enumeration of several kinds of peaks.…”

Section: Introductionmentioning

confidence: 99%

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Symmetric and asymmetric peaks or valleys in (partial) Dyck paths

Sun¹,

Shi²,

Zhao³

2021

Preprint

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The concepts of symmetric and asymmetric peaks in Dyck paths were introduced by Flórez and Ramírez, who counted the total number of such peaks over all Dyck paths of a given length. Elizalde generalized their results by giving multivariate generating functions that keep track of the number of symmetric peaks and the number of asymmetric peaks. Elizalde also considered the analogous notion of symmetric valleys by a continued fraction method. In this paper, mainly by bijective methods, we devote to enumerating the statistics "symmetric peaks", "asymmetric peaks", "symmetric valleys" and "asymmetric valleys" of weight k + 1 over all (partial) Dyck paths of a given length. Our results refine some consequences of Flórez and Ramírez, and Elizalde.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symmetric and asymmetric peaks or valleys in (partial) Dyck paths

Sun¹,

Shi²,

Zhao³

2021

Preprint

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“…For a detailed study of properties of Dyck paths see [6]. For further recent work on Dyck paths, see [1,2,3,4,5,8,14].…”

Section: General Introductionmentioning

confidence: 99%

Left to right maxima in Dyck Paths

Blecher¹,

Knopfmacher²

2021

Preprint

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“…, 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%

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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Nonleft peaks in Dyck paths: A combinatorial approach

Cited by 6 publications

References 8 publications

Symmetric and asymmetric peaks or valleys in (partial) Dyck paths

Symmetric and asymmetric peaks or valleys in (partial) Dyck paths

Left to right maxima in Dyck Paths

Restricted Dyck Paths on Valleys Sequence

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