Related papers: Rook and queen paths with boundaries
Evaluation of regular path queries (RPQs) is a central problem in graph databases. We investigate the corresponding enumeration problem, that is, given a graph and an RPQ, enumerate all paths in the graph that match the RPQ. We consider…
We consider a discrete random walk on a diagonal lattice in two and three dimensions and obtain explicit solutions of absorption probabilities and probabilities of return in several domains. In three dimensions we consider both the cube and…
The Catalan numbers count many classes of combinatorial objects. The most emblematic such objects are probably the Dyck walks and the binary trees, and, whenever another class of combinatorial objects is counted by the Catalan numbers, it…
We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time…
There was recent interest in Motzkin paths without peaks (peak: up-step followed immediately by down-step); additional results about this interesting family is worked out. The new results are the enumeration of such paths that live in a…
The Gessel number $P(n,r)$ represents the number of lattice paths in a plane with unit horizontal and vertical steps from $(0,0)$ to $(n+r,n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x \geq r\}$. In…
A path system $\mathscr{P}$ in a graph $G=(V,E)$ is a collection of paths, with exactly one path between any two vertices in $V$. A path system is said to be consistent if it is closed under subpaths. We say that a path system $\mathscr{P}$…
A plateau in a Motzkin path is a sequence of three steps: an up step, a horizontal step, then a down step. We find three different forms for the bivariate generating function for plateaus in Motzkin paths, then generalize to longer…
A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and has a trapped endpoint. We show that the generating function enumerating GSAWs on a half-infinite strip of finite height is…
We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The non-holonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence…
Given a coprime pair $(m,n)$ of positive integers, rational Catalan numbers $\frac{1}{m+n} \binom{m+n}{m,n}$ counts two combinatorial objects:rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the…
Nested parentheses are forms in an algebra which define orders of evaluations. A class of well-formed sets of associated opening and closing parentheses is well studied in conjunction with Dyck paths and Catalan numbers. Nested parentheses…
We analyze the combinatorics behind the operation of taking the logarithm of the generating function $G_k$ for $k^\text{th}$ generalized Catalan numbers. We provide combinatorial interpretations in terms of lattice paths and in terms of…
We introduce the algorithmic problem of finding a locally rainbow path of length $\ell$ connecting two distinguished vertices $s$ and $t$ in a vertex-colored directed graph. Herein, a path is locally rainbow if between any two visits of…
In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the…
Cops and robbers is a pursuit-evasion game played on graphs. We completely classify the cop numbers for $n \times n$ knight graphs and queen graphs. This completes the classification of the cop numbers for all $n \times n$ classical chess…
We consider a simple symmetric random walk on a spider, that is a collection of half lines (we call them legs) joined at the origin. Our main question is the following: if the walker makes $n$ steps how high can he go up on all legs. This…
The problem of a restricted random walk on graphs which keeps track of the number of immediate reversal steps is considered by using a transfer matrix formulation. A closed-form expression is obtained for the generating function of the…
We present an analytical approach to study simple symmetric random walks (RWs) on a crossing geometry consisting of a plane square lattice crossed by $n_l$ number of lines that all meet each other at a single point (the origin) on the…
A rotor-router walk on a graph is a deterministic process, in which each vertex is endowed with a rotor that points to one of the neighbors. A particle located at some vertex first rotates the rotor in a prescribed order, and then it is…