The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length n with flaws m is the n-th Catalan number and independent on m. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this paper, we find the connections between these two Chung-Feller theorems. We focus on the weighted versions of three classes of lattice paths and give the generalizati...

A binary string B of length n = kt is a k-ary Dyck word if it contains t copies of 1, and the number of 0s in every prefix of B is at most k−1 times the number of 1s. We provide two loopless algorithms for generating k-ary Dyck words in cool-lex order: (1) The first requires two index variables and assumes k is a constant; (2) The second requires t index variables and works for any k. We also e...

A Dyck path is a lattice path in the plane integer lattice Z × Z consisting of steps (1, 1) and (1,−1), which never passes below the x-axis. A peak at height k on a Dyck path is a point on the path with coordinate y = k that is immediately preceded by a (1, 1) step and immediately followed by a (1,−1) step. In this paper we find an explicit expression to the generating function for the number o...

A Dyck path is a lattice path in the plane integer lattice Z × Z consisting of steps (1, 1) and (1,−1), each connecting diagonal lattice points, which never passes below the x-axis. The number of all Dyck paths that start at (0, 0) and finish at (2n, 0) is also known as the nth Catalan number. In this paper we find a closed formula, depending on a non-negative integer t and on two lattice point...

Leapfrog transformations starting with the genus 3 Klein and Dyck tessellations consisting of 24 heptagons and 12 octagons, respectively, can generate possible highly symmetrical structures for allotropes of carbon and the isosteric boron nitride, (BN)x. The Klein tessellation, alternatively described as a platonic 3,7 tessellation, corresponds to the Riemann surface for the multi-valued functi...

Model-based intrusion detection compares a process’s execution against a program model to detect intrusion attempts. Models constructed from static program analysis have historically traded precision for efficiency. We address this problem with our Dyck model, the first efficient statically-constructed context-sensitive model. This model specifies both the correct sequences of system calls that...

A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) consisting of upsteps 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 genera...

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n+1,C). We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals an...

Abstract. 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-segme...