Related papers: Peakless Motzkin paths of bounded height

We introduce a new concept of permutation avoidance pattern called hatted pattern, which is a natural generalization of the barred pattern. We show the growth rate of the class of permutations avoiding a hatted pattern in comparison to…

A condition is given, under which a general lattice point counting function is asymptotic to the corresponding ball volume growth function. This is then used to give height asymptotics in the style of the Batyrev-Manin Conjecture for…

In a Dyck path a peak which is (weakly) higher than all the preceding peaks is called a strict (weak) left to right maximum. We obtain explicit generating functions for both weak and strict left to right maxima in Dyck paths. The proofs of…

We consider Motzkin paths of length $L$, not fixed at zero at both end points, with constant weights on the edges and general weights on the end points. We investigate, as the length $L$ tends to infinity, the limit behaviors of (a)…

In this paper, we consider $N$ non-intersecting Bessel paths starting at $x=a\geq 0$, and conditioned to end at the origin $x=0$. We derive the explicit formula of the distribution function for the maximum height. Depending on the starting…

We relate the combinatorics of periodic generalized Dyck and Motzkin paths to the cluster coefficients of particles obeying generalized exclusion statistics, and obtain explicit expressions for the counting of paths with a fixed number of…

A well-known bijection between Motzkin paths and ordered trees with outdegree always $\le2$, is lifted to Grand Motzkin paths (the nonnegativity is dropped) and an ordered list of an odd number of such $\{0,1,2\}$ trees. This offers an…

We investigate what happens when an entire sample path of a smooth Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high…

We consider the height of random k-trees and k-Apollonian networks. These random graphs are not really trees, but instead have a tree-like structure. The height will be the maximum distance of a vertex from the root. We show that w.h.p. the…

Motzkin paths of order-$\ell$ are a generalization of Motzkin paths that use steps $U=(1,1)$, $L=(1,0)$, and $D_i=(1,-i)$ for every positive integer $i \leq \ell$. We further generalize order-$\ell$ Motzkin paths by allowing for various…

In this paper we find recurrence relations for the asymptotic probability a vertex is $k$ protected in all Motzkin trees. We use a similar technique to calculate the probabilities for balanced vertices of rank $k$. From this we calculate…

Let $G$ be a directed graph on finitely many vertices and edges, and assign a positive weight to each edge on $G$. Fix vertices $u$ and $v$ and consider the set of paths that start at $u$ and end at $v$, self-intersecting in any number of…

We investigate paths in the hexagonal circle packing and enumerate them with respect to width, height, number of steps, area, and kissing number. Functional equations and the kernel method yield closed bivariate generating functions…

Skew Dyck paths are a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ a south-west step $(-1,-1)$ is also allowed, provided that the path does not intersect itself. Replacing the south-west step by a red south-east…

We consider a constrained version of the shortest path problem on the complete graphs whose edges have independent random lengths and costs. We establish the asymptotic value of the minimum length as a function of the cost-budget within a…

We study the enumeration of different classes of grand knight's paths in the plane. In particular, we focus on the subsets of zigzag knight's paths that are subject to constraints. These constraints include ending at $y$-coordinate 0,…

We provide formulas for generating functions of many types of paths in various rooted tree structures. We compute the $k$th moment of the generating functions for various types of vertical paths. In two specific familes of trees we find…

Analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found. We apply the approach to classical random graphs of Erdos and Renyi (ER), evolving networks introduced by…

A dispersed Dyck path (DDP) of length n is a lattice path on $N\times N$ from (0,0) to (n,0) in which the following steps are allowed: "up" (x, y) $\to$ (x+1, y+1); "down" (x, y) $\to$ (x+1, y-1); and "right" (x,0) $\to$ (x+1,0). An ascent…

Valleyless sequences of finite length $n$ and maximum entry $k$ occur in tree enumeration problems and provide an interesting correspondence between permutations and compositions. In this paper we introduce the notion of \emph {valleyless}…