Enumerations of peaks and valleys on non-decreasing Dyck paths

Czabarka, Éva; Flórez, Rigoberto; Junes, Leandro; Ramírez, José L.

doi:10.1016/j.disc.2018.06.032

Cited by 14 publications

(13 citation statements)

References 6 publications

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“…Notice, a property that is true in this generalization, immediately, holds for non-decreasing Dyck paths (taking t ¼ 2). In particular, we extend several results from [2,7,11]. As a result, we have found that the generating functions, of the aspects counted here, are rational.…”

Section: Introductionsupporting

confidence: 78%

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Enumerations of Rational Non-decreasing Dyck Paths with Integer Slope

Flórez

Ramírez

2021

Graphs and Combinatorics

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We extend the concept of non-decreasing Dyck paths to t-Dyck paths. We denote the set of non-decreasing t-Dyck paths by D t . Several classic questions studied in other families of lattice paths are studied here for D t . We use generating functions, recursive relations and Riordan arrays to count, for example, the following aspects: the number of non-decreasing paths in D t with a given fixed length, the total number of prefixes of all paths in D t of a given length, and the total number of paths in D t with a fixed number of peaks. We give a generating function to count the number of paths in D t that can be written as a concatenation of a given fixed number of primitive paths and we give a relation between paths in D t and direct columnconvex polyominoes.

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Section: Introductionsupporting

confidence: 78%

“…In 1997 Barcucci et al [2] introduced the concept of non-decreasing Dyck paths, and since then many papers, studying several aspects of this family of paths, have appeared [2,3,[6][7][8][9][10][11][12][13]19]. A t-Dyck path is a generalization of the Dyck path concept where every South-East step has length of ðt À 1Þ.…”

Section: Introductionmentioning

confidence: 99%

Enumerations of Rational Non-decreasing Dyck Paths with Integer Slope

Flórez

Ramírez

2021

Graphs and Combinatorics

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“…Note that every low peak is symmetric. By [6,Corollary 6], the total number of high peaks in D n is ((2n − 1)F 2n − nF 2n−1 )/5. Together with Corollary 2.5, this implies the following.…”

Section: Recurrence Relations and Fibonacci Numbersmentioning

confidence: 99%

“…A Dyck path is called non-decreasing if the heights of its valleys form a non-decreasing sequence from left to right (see Figure 1 for an example). Non-decreasing Dyck paths have been extensively studied in the literature, see [2,5,6,8,14,16,17,20]. All the Dyck paths considered in this paper will be non-decreasing.…”

Section: Introductionmentioning

confidence: 99%

“…All the Dyck paths considered in this paper will be non-decreasing. Following the notation from [5,6,13,14], we denote by D the set of all non-decreasing Dyck paths, and by D n the set of all non-decreasing Dyck paths of length 2n, where the length is defined as the number of steps. For P ∈ D n , we write |P |= n to denote its semilength.…”

Section: Introductionmentioning

confidence: 99%

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Enumerating symmetric peaks in non-decreasing Dyck paths

2021

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Local maxima and minima of a Dyck path are called peaks and valleys, respectively. A Dyck path is non-decreasing if the heights (y-coordinates) of its valleys increase from left to right. A peak is symmetric if it is surrounded by two valleys (or endpoints of the path) at the same height. In this paper we give multivariate generating functions, recurrence relations, and closed formulas to count the number of symmetric and asymmetric peaks in non-decreasing Dyck paths. Finally, we use Riordan arrays to study weakly symmetric peaks, namely those for which the valley preceding the peak is at least as high as the valley following it.

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