Related papers: On k-Dyck paths with a negative boundary
In an effort to further understanding $q,t$-Catalan statistics, a new statistic on Dyck paths called $\mathtt{depth}$ was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known…
We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all…
This note deals with the relationship between the total number of $k$-walks in a graph, and the sum of the $k$-th powers of its vertex degrees. In particular, it is shown that the the number of all $k$-walks is upper bounded by the sum of…
We enumerate the number of monotonic lattice paths starting at $(0,0)$ and terminating at $(m,n)$ in which $l$ of the first $k$ steps lie below the line $y=x\ (0\leq k\leq m\leq n)$. These closed formulas consist of terms which are a…
Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which…
This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…
We construct examples of a random walk with pairwise-independent steps which is almost-surely bounded, and for any $m$ and $k$ a random walk with $k$-wise independent steps which has no stationary distribution modulo $m$.
Recently, Mansour and Shattuck related the total number of humps in all of the $(k, a)$-paths of order $n$ to the number of super $(k, a)$-paths, which generalized previous results concerning the cases when $k = 1$ and $a = 1$ or $a =…
Stanley considered Dyck paths where each maximal run of down-steps to the $x$-axis has odd length; they are also enumerated by (shifted) Catalan numbers. Prefixes of these combinatorial objects are enumerated using the kernel method. A more…
New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…
We present a new bijection between variants of $m$-Dyck paths (paths with steps in $\{+1,-m\}$ starting and ending at height $0$ and remaining at non-negative height), which generalizes a classical bijection between Dyck prefixes and…
We exhibit a bijection between central Delannoy $n$-paths, that is, lattice paths from the origin to $(n,n)$ with steps $E=(1,0), \,N=(0,1),\,D=(1,1)$ and the lattice paths from the origin to $(n+1,n)$ where the only restriction on the…
The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from $(0,0)$ to $(n,n)$ which is below the diagonal line $y=x$. One way to generalize the definition of Dyck path is…
There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the…
An oriented graph is called $k$-anti-traceable if the subdigraph induced by every subset with $k$ vertices has a hamiltonian anti-directed path. In this paper, we consider an anti-traceability conjecture. In particular, we confirm this…
We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we…
Descents of odd length in Dyck paths are discussed, taking care of some variations. The approach is based on generating functions and the kernel method and augments relations about them from the Encyclopedia of Integer Sequences, that were…
It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fu{\ss}-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps.…
For an integer $k\ge 3$, a $k$-path vertex cover of a graph $G=(V,E)$ is a set $T\subseteq V$ that shares a vertex with every path subgraph of order $k$ in $G$. The minimum cardinality of a $k$-path vertex cover is denoted by $\psi_k(G)$.…
The paper is devoted to the study of lattice paths that consist of vertical steps $(0,-1)$ and non-vertical steps $(1,k)$ for some $k\in \mathbb Z$. Two special families of primary and free lattice paths with vertical steps are considered.…