Standard Young tableaux and colored Motzkin paths

Eu, Sen Peng; Fu, Tung-Shan; Hou, Justin T.; Hsu, Te Wei

doi:10.1016/j.jcta.2013.06.007

Cited by 15 publications

(16 citation statements)

References 9 publications

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“…This leaves open the possibility that other algorithms could be described, which give the same output but a different pairing. In fact, in [9], Eu, Fu, Hou, and Hsu give a description of the bijection by an algorithm giving a different pairing. See also [8].…”

Section: Freedom Of Pairing and Comparison With Eu Fu Hou And Hsumentioning

confidence: 99%

“…While the bijection suggested by Gouyou-Beauchamps [11] proceeds by composing several known bijections, more recent works by Eu and collaborators [8,9] introduced a more direct transformation that operates by searching for patterns on the walk viewed as a word and successively rewriting their occurrences until none is found. The patterns used are arbitrary distant pairs of matching parentheses.…”

Section: Introductionmentioning

confidence: 99%

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Bijections between Łukasiewicz Walks and Generalized Tandem Walks

Chyzak

Yeats

2020

Electron. J. Combin.

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In this article, we study the enumeration by length of several walk models on the square lattice. We obtain bijections between walks in the upper half-plane returning to the x-axis and walks in the quarter plane. A recent work by Bostan, Chyzak, and Mahboubi has given a bijection for models using small north, west, and south-east steps. We adapt and generalize it to a bijection between half-plane walks using those three steps in two colours and a quarterplane model over the symmetrized step set consisting of north, north-west, west, south, south-east, and east. We then generalize our bijections to certain models with large steps: for given p ≥ 1, a bijection is given between the half-plane and quarter-plane models obtained by keeping the small south-east step and replacing the two steps north and west of length 1 by the p + 1 steps of length p in directions between north and west. This model is close to, but distinct from, the model of generalized tandem walks studied by Bousquet-Mélou, Fusy, and Raschel.1 In the absence of a written text yet, one can consult [7] for an earlier, related presentation. 2 The original presentation in [3] uses automata with infinitely many states and no auxiliary memory, but, in view of the generalization to come, we prefer an equivalent presentation with finitely many states and counters that encode the numbering of the states considered in [3].-Tandem words are classically those words on S 1 giving rise to a walk in the quarter plane. We will also call them quarter-plane 1-tandem words:-By relaxing the quarter-plane constraint, but forcing the final height, we obtain what we will call half-plane 1-tandem words:The bijection M H is obvious.• {1, 2, 3}. We restrict the classical class of Yamanouchi words (using any letters from N * ) to those words using only the first three integers:Y 3 = {w ∈ {1, 2, 3} * : ∀w , w ≤ w ⇒ |w | 1 ≥ |w | 2 ≥ |w | 3 }.The bijection Y 3 Q is obvious.

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Section: Freedom Of Pairing and Comparison With Eu Fu Hou And Hsumentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bijections between Łukasiewicz Walks and Generalized Tandem Walks

Chyzak

Yeats

2020

Electron. J. Combin.

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“…In Zeilberger's lazy walks, the k = 1 case encodes the classic Motzkin walks, consistent with the longstanding observation that y 3 (n) is the number of Motzkin words. One could view the higher dimension lazy walks as a generalization of Motzkin words, and this notion was first formalized by Eu [14], and subsequently by Eu, Fu, Hou, and Hsu [15]. They add a counter component, and describe an explicit bijection between the Motzkin paths of length n and the standard Young tableaux of size n with at most three rows.…”

Section: Lazy Walksmentioning

confidence: 99%

“…Bijective proofs of some of these connections are more recent [15,9,28], although the work of Gouyou-Beauchamps dates back to the late 1980s. All of these authors' proofs pass through secondary objects, such as coloured Motzkin paths, or matchings.…”

Section: Introductionmentioning

confidence: 99%

On Standard Young Tableaux of Bounded Height

Mishna

2019

Association for Women in Mathematics Series

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“…(c) Yamanouchi-colored Motzkin paths, for which a different bijection to the one we use is given by Eu et al [10]. We conclude in Section 7 with several remarks.…”

Section: Introductionmentioning

confidence: 99%

From Dyck Paths to Standard Young Tableaux

et al. 2020

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We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength n that is shown to be in bijection with the set of all SYT with n boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some sense is dual to the known class of noncrossing partitions.2010 Mathematics Subject Classification. 05A19 (Primary); 05A05 (Secondary).

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Standard Young tableaux and colored Motzkin paths

Cited by 15 publications

References 9 publications

Bijections between Łukasiewicz Walks and Generalized Tandem Walks

Bijections between Łukasiewicz Walks and Generalized Tandem Walks

On Standard Young Tableaux of Bounded Height

From Dyck Paths to Standard Young Tableaux

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