相关论文: Short note on the number of 1-ascents in dispersed…
Motzkin paths consist of up-steps, down-steps, level-steps, and never go below the $x$-axis. They return to the $x$-axis at the end. The concept of skew Dyck path \cite{Deutsch-italy} is transferred to skew Motzkin paths, namely, a left…
We consider Dyck paths having height at most two with some constraints on the number of consecutive valleys at height one which must be followed by a suitable number of valleys at height zero. We prove that they are enumerated by so-called…
The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size -3,-5,-7,... . For such paths, we find the generating functions of them, according to length, ending at level $i$,…
We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.
Skew Dyck paths are like Dyck paths, but an additional south-west step $(-1,-1)$ is allowed, provided that the path does not intersect itself. Lattice paths with catastrophes can drop from any level to the origin in just one step. We…
We study the expected distance of short uniform random walks in arbitrary dimensions with unit steps in random directions. It is known that for dimensions $d=2$ and $d=4$, all the moments of an $m$-step walk are integer. While for $d=2$,…
Motzkin paths with air pockets (MAP) are defined as a generalization of Dyck paths with air pockets by adding some horizontal steps with certain conditions. In this paper, we introduce two generalizations. The first one consists of lattice…
The degree of symmetry of a combinatorial object, such as a lattice path, is a measure of how symmetric the object is. It typically ranges from zero, if the object is completely asymmetric, to its size, if it is completely symmetric. We…
We compute the limiting distribution of height of a random discrete excursion with step sets consisting of one positive step 1 and arbitrary finite set of non-positive integers. The limit law is the supremum of a Brownian excursion. This is…
We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form for $A(x)=\sum_{k\geq 0}A_k(x)$. We apply…
Skew Dyck 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 step,…
The enumeration of k-Dyck paths ending at level j after m up-steps, where the last step is an up-step, is given as a sum, improving on a previous formula given by Deng and Mansour.
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…
Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in $\mathbb{N}^2$ starting at the origin made of steps $U=(1,1)$, $D_k=(1,-k)$, $k\geq 1$ and…
We interpret walks in the first quadrant with steps {(1,1),(1,0),(-1,0), (-1,-1)} as a generalization of Dyck words with two sets of letters. Using this language, we give a formal expression for the number of walks in the steps above…
The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths:…
This work presents new asymptotic formulas for family of walks in Weyl chambers. The models studied here are defined by step sets which exhibit many symmetries and are restricted to the first orthant. The resulting formulas are very…
A constrained diffusive random walk of n steps and a random flight in Rd, which can be expressed in the same terms, were investigated independently in recent papers. The n steps of the walk are identically and independently distributed…
Dyck paths where peaks are only allowed on level 1 and on even-indexed levels, were introduced by Retakh and analysed by Zeilberger, with assistance from Ekhad. We add some combinatorial comments to the enumeration, which involves Motzkin…
A 1d random geometric graph (1d RGG) is built by joining a random sample of $n$ points from an interval of the real line with probability $p$. We count the number of $k$-hop paths between two vertices of the graph in the case where the…