Counting peaks at height k in a Dyck path (original) (raw)

Counting Dyck Paths According to the Maximum Distance Between Peaks and Valleys

Journal of Integer Sequences

A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) consisting of up- steps u = (1, 1) and down-steps d = (1, 1) which never passes below the x-axis. Let Dn denote the set of Dyck paths of length 2n. A peak is an occurrence of ud (an upstep immediately followed by a downstep) within a Dyck path, while a valley is an occurrence of du. Here, we compute explicit formulas for the generating functions which count the members of Dn according to the maximum number of steps between any two peaks, any two valleys, or a peak and a valley. In addition, we provide closed expressions for the total value of the corresponding statistics taken over all of the members of Dn. Equivalent statistics on the set of 231-avoiding permutations of length n are also described.

Bell polynomials and -generalized Dyck paths

Discrete Applied Mathematics, 2008

A k-generalized Dyck path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z consisting of horizontal-steps (k, 0) for a given integer k ≥ 0, up-steps (1, 1), and down-steps (1, −1), which never passes below the x-axis. The present paper studies three kinds of statistics on k-generalized Dyck paths: "number of u-segments", "number of internal u-segments" and "number of (u, h)-segments". The Lagrange inversion formula is used to represent the generating function for the number of k-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to u-segments and (u, h)-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers.

Returns and Hills on Generalized Dyck Paths

J. Integer Seq., 2016

In 2009, Shapiro posed the following question: “What is the asymptotic proportion of Dyck paths having an even number of hills?” In this paper, we answer Shapiro’s question, as well as a generalization of the question to ternary paths. We find that the probability that a randomly chosen ternary path has an even number of hills approaches 125/169 as the length of the path approaches infinity. Our strategy relies on properties of the Fine number sequence and extends certain relationships between the Catalan and Fine number generating functions.

Explicit Formulas for Enumeration of Lattice Paths: Basketball and the Kernel Method

Lattice Path Combinatorics and Applications, 2019

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude j and ending at a given altitude k, with additional constraints such as, for example, to never attain altitude 0 in-between. We first discuss the case of walks on the integers with steps −h,. .. , −1, +1,. .. , +h. The case h = 1 is equivalent to the classical Dyck paths, for which many ways of getting explicit formulas involving Catalan-like numbers are known. The case h = 2 corresponds to "basketball" walks, which we treat in full detail. Then we move on to the more general case of walks with any finite set of steps, also allowing some weights/probabilities associated with each step. We show how a method of wide applicability, the so-called "kernel method", leads to explicit formulas for the number of walks of length n, for any h, in terms of nested sums of binomials. We finally relate some special cases to other combinatorial problems, or to problems arising in queuing theory.

Enumeration of Colored Dyck Paths Via Partial Bell Polynomials

Developments in Mathematics, 2019

We consider a class of lattice paths with certain restrictions on their ascents and down steps and use them as building blocks to construct various families of Dyck paths. We let every building block Pj take on cj colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in an unified manner.

Enumeration of Dyck paths with air pockets

2022

We introduce and study the new combinatorial class of Dyck paths with air pockets. We exhibit a bijection with the peakless Motzkin paths which transports several pattern statistics and give bivariate generating functions for the distribution of patterns as peaks, returns and pyramids. Then, we deduce the popularities of these patterns and point out a link between the popularity of pyramids and a special kind of closed smooth self-overlapping curves, a subset of Fibonacci meanders. A similar study is conducted for the subclass of non-decreasing Dyck paths with air pockets.

Restricted Dyck Paths on Valleys Sequence

2021

Abstract. 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.