Skew Dyck paths

Deutsch, Emeric; Munarini, Emanuele; Rinaldi, Simone

doi:10.1016/j.jspi.2010.01.015

Cited by 19 publications

(16 citation statements)

References 11 publications

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“…Then we move to skew Dyck paths [9]. They are like Dyck paths, but allow for an extra step (−1, −1), provided that the path does not intersect itself.…”

Section: Hoppy Walksmentioning

confidence: 99%

“…They are like Dyck paths, but allow for an extra step (−1, −1), provided that the path does not intersect itself. An equivalent model, defined and described using a bijection, is from [9]: Marked ordered trees. They are like ordered trees, with an additional feature, namely each rightmost edge (except for one that leads to a leaf) can be coloured with two colours.…”

Section: Hoppy Walksmentioning

confidence: 99%

“…In [9] we find the following variation of ordered trees: Each rightmost edge might be marked or not, if it does not lead to an endnode (leaf). We depict a marked edge by the red colour and draw all of them of size 4 (4 nodes):…”

Section: Marked Ordered Treesmentioning

confidence: 99%

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A walk in my lattice path garden

Prodinger¹

2021

Preprint

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Various lattice path models are reviewed. The enumeration is done using generating functions. A few bijective considerations are woven in as well. The kernel method is often used. Computer algebra was an essential tool. Some results are new, some have appeared before.The lattice path models we treated, are: Hoppy's walks, the combinatorics of sequence A002212 in [47] (skew Dyck paths, Schröder paths, Hex-trees, decorated ordered trees, multi-edge trees, etc.) Weighted unary-binary trees also occur, and we could improve on our old paper on Horton-Strahler numbers [17], by using a different substitution. Some material on ternary trees appears as well, as on Motzkin numbers and paths (a model due to Retakh), and a new concept called amplitude that was found in [6]. Some new results on Deutsch paths in a strip are included as well. During the Covid period, I spent much time with this beautiful concept that I dare to call Deutsch paths, since Emeric Deutsch stands at the beginning with a problem that he posted in the American Mathematical Monthly some 20 years ago. Peaks and valleys, studied by Rainer Kemp 40 years under the names max-turns and min-turns, are revisited with a more modern approach, streamlining the analysis, relying on the 'subcritical case' (named so by Philippe Flajolet), the adding a new slice technique and once again the kernel method.

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“…Then we move to skew Dyck paths [9]. They are like Dyck paths, but allow for an extra step (−1, −1), provided that the path does not intersect itself.…”

Section: Hoppy Walksmentioning

confidence: 99%

Section: Hoppy Walksmentioning

confidence: 99%

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A walk in my lattice path garden

Prodinger¹

2021

Preprint

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“…Figures 12 (c),(d) show two different examples of paths confined to C that are not steady paths. The reader may observe that steady paths can be regarded as a subfamily of those "skew Dyck paths" considered in [19] and enumerated according to several parameters. (c) a path in C that violates (S1); (d) a path in C that violates (S2).…”

Section: Definition and Succession Rulementioning

confidence: 99%

Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers

Beaton¹,

Bouvel²,

Guerrini³

et al. 2019

Theoretical Computer Science

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The first problem addressed by this article is the enumeration of some families of patternavoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Lin, and Kim and Lin. The strength of our approach is its robustness: we enumerate four families F1 ⊂ F2 ⊂ F3 ⊂ F4 of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family Fi which generalizes the one for the family Fi−1.The second topic of the paper is the enumeration of a fifth family F5 of pattern-avoiding inversion sequences (containing F4). This enumeration is also solved via a succession rule, which however does not generalize the one for F4. The associated enumeration sequence, which we call the powered Catalan numbers, is quite intriguing, and further investigated. We provide two different succession rules for it, denoted ΩpCat and Ω steady , and show that they define two types of families enumerated by powered Catalan numbers. Among such families, we introduce the steady paths, which are naturally associated with Ω steady . They allow us to bridge the gap between the two types of families enumerated by powered Catalan numbers: indeed, we provide a size-preserving bijection between steady paths and valley-marked Dyck paths (which are naturally associated with ΩpCat).Along the way, we provide several nice connections to families of permutations defined by the avoidance of vincular patterns, and some enumerative conjectures. arXiv:1808.04114v2 [math.CO] 14 Dec 2018 Proposition 2. Any inversion sequence e = (e 1 , . . . , e n ) is a Catalan inversion sequence if and only if for any i, with 1 ≤ i < n, if e i forms a weak descent, i.e. e i ≥ e i+1 , then e i < e j , for all j > i + 1.Proof. The forward direction is clear. The backwards direction can be proved by contrapositive. More precisely, suppose there are three indices i < j < k, such that e i ≥ e j , e k . Then, if e j = e i+1 , e i forms a weak descent and the fact that e i ≥ e k concludes the proof. Otherwise, since e i ≥ e j , there must be an index i , with i ≤ i < j, such that e i forms a weak descent and e i ≥ e k . This concludes the proof as well.

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“…then the sequence A002212 is obtained, some properties about these numbers are established in [9,24]. If k = 4 the sequence A005572 which starts 1,4,17,76,354,1704,8421, 42508 is obtained.…”

Section: Proof From Theorem 24 the Gf Ismentioning

confidence: 99%

Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs

Castro¹,

Ramírez²,

Ramírez³

2014

SACS

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In this paper we present a general methodology to solve a wide variety of classical lattice path counting problems in an uniform way. These counting problems are related to Dyck paths, Motzkin paths and some generalizations. The methodology uses weighted automata, equations of ordinary generating functions and continued fractions. It is a variation of the technique proposed by J. Rutten, which is called Coinductive Counting.

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Skew Dyck paths

Cited by 19 publications

References 11 publications

A walk in my lattice path garden

A walk in my lattice path garden

Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers

Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs

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