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The Statistic “Number of Udu's” in Dyck Paths
Discrete Mathematics, 2004 This paper studies the enumeration of Dyck paths by the number of \(udu\)'s (up-down-up). More precisely, the enumeration of Dyck paths with given semilength and given number of \(udu\)'s is obtained. An \(udu\) is said to be at low (high) level, if its peak is at level (greater than) one.
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On Rational Dyck Paths and the Enumeration of Factor-Free Dyck Words [PDF]
Discrete Applied Mathematics, 2018 10 pages, 2 figures.
Daniel Birmajer, Michael D Weiner
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A refinement of Dyck paths: A combinatorial approach
Discrete Mathematics, Algorithms and Applications, 2021Local maxima and minima of a Dyck path are called peaks and valleys, respectively. A Dyck path is called restricted [Formula: see text]-Dyck if the difference between any two consecutive valleys is at least [Formula: see text] (right-hand side minus left-hand side) or if it has at most one valley.
Florez, Rigoberto +4 more
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Journal of Statistical Planning and Inference, 2010
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Emanuele Munarini +2 more
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zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Emanuele Munarini +2 more
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Visits to Level r by Dyck Paths
Fundamenta Informaticae, 2012A Dyck path is a non-negative lattice path in $\mathbb{N}^2$ starting at the origin, where only two types of steps are allowed: the diagonal up step (1, 1) and the diagonal down step (1, −1). The length of the path is the total number of unit steps. We consider paths of length n, ending at the point (n, i).
Charlotte A. C. Brennan, Simon Mavhungu
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Area and Inertial Moment of Dyck Paths
Combinatorics, Probability and Computing, 2004In this paper, we investigate the limit law of the inertial moment of Dyck paths with respect to the $x$-axis, that is, the sum of the squares of the altitudes. We find its Laplace transform using Louchard's methodology, rediscovering a result which was in fact well known by probabilists.
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Efficient exact paths for dyck and semi-dyck labeled path reachability (extended abstract)
2017 IEEE 8th Annual Ubiquitous Computing, Electronics and Mobile Communication Conference (UEMCON), 2017Consider any two vertices in a weighted digraph. The exact path length problem is to determine if there is a path of a given fixed cost between these vertices. This paper focuses on the exact path problem for costs −1,0 or +1 between all pairs of vertices. This special case is also restricted to original edge weights from {−1, +1}. In this special case,
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Refinements of (\(n,m\))-Dyck paths
Eur. J. Comb., 2011A \((n,m)\)-Dyck path is a lattice path in \(\mathbb{Z}\times\mathbb{Z}\) using up \((1,1)\) and down \((1,-1)\) steps that go from the origin to the point \((2n,0)\) and it contains exactly \(m\) up steps under the \(x\)-axis. The classical Chung-Feller theorem tells us that the number of \((n,m)\)-Dyck paths is \(\frac{1}{n+1}\binom{2n}{n}\) the \(n\)
Jun Ma 0017, Yeong-Nan Yeh
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Returns and hills on generalized Dyck paths
J. Integer Seq., 2016Summary: In 2009, Shapiro posed the following question: ``What is the asymptotic proportion of Dyck paths having an even number of hills?'' (see [\textit{G.-S. Cheon} et al., Eur. J. Comb. 31, No. 1, 120--128 (2010; Zbl 1184.05006)]). In this paper, we answer Shapiro's question, as well as a generalization of the question to ternary paths. We find that
Naiomi T. Cameron, Jillian E. McLeod
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Partial Skew Dyck Paths: A Kernel Method Approach
Graphs and Combinatorics, 2022Helmut Prodinger
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