Related papers: Non-decreasing Deutsch paths
For any pattern $p$ of length at most two, we provide generating functions and asymptotic approximations for the number of $p$-equivalence classes of Dyck paths with catastrophes, where two paths of the same length are $p$-equivalent…
A special case of a conjecture by Thomass\'e is that any oriented graph with minimum outdegree k contains a dipath of length 2k. For the sake of proving whether or not a counterexample exists, we present reductions and establish bounds on…
\L{}ukasiewicz paths are lattice paths in $\Bbb{N}^2$ starting at the origin, ending on the $x$-axis, and consisting of steps in the set $\{(1,k), k\geq -1\}$. We give generating function and exact value for the number of $n$-length…
We introduce the notion of a "walk with jumps", which we conceive as an evolving process in which a point moves in a space (for us, typically $\mathbb{H}^2$) over time, in a consistent direction and at a consistent speed except that it is…
Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths, in which tiles are no larger than the tiles they cover. These tilings arise in the study of certain statistical physics models and…
Let $(Z_n)_{n\in\N}$ be a $d$-dimensional {\it random walk in random scenery}, i.e., $Z_n=\sum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\in\N_0}$ a random walk in $\Z^d$ and $(Y(z))_{z\in\Z^d}$ an i.i.d. scenery, independent of the walk. The…
We are given an acyclic directed graph with one source, and a subset of its edges which contains exactly one outgoing edge for every non-sink vertex. These edges determine a unique path from the source to a sink. We can think of it as a…
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…
Stanley lists the class of Dyck paths where all returns to the axis are of odd length as one of the many objects enumerated by (shifted) Catalan numbers. By the standard bijection in this context, these special Dyck paths correspond to a…
Dynamic systems characterized by diversified evolutions are not only more flexible, but also more resilient to attacks, failures and changing conditions. This article addresses the quantification of the diversity of non-linear transient…
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.…
We consider a class of lattice paths with certain restrictions on their ascents and down steps and use them as building blocks to construct various families of Dyck paths. We let every building block $P_j$ take on $c_j$ colors and count all…
We show bijectively that the Catalan number C_n counts Dyck (n+1)-paths in which the terminal descent is of even length and all other descents to ground level (if any) are of odd length.
We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically…
In this paper we study two types of descent in the category of Berkovich analytic spaces: flat descent and descent with respect to an extension of the ground field. Quite surprisingly, the deepest results in this direction seem to be of the…
We exploit Krattenthaler's bijection between the set $S_n(3\textrm{-}1\textrm{-}2)$ of permutations in $S_n$ avoiding the classical pattern $3\textrm{-}1\textrm{-}2$ and Dyck $n$-paths to study the distribution of every consecutive pattern…
Pappe, Paul, and Schilling introduced two combinatorial statistics, depth and ddinv, associated with classical Dyck paths, and proved that the distributions of (area, depth) and (dinv, ddinv) are $q,t$-symmetric by constructing an…
We introduce the classes of descendingly flexible and descendingly alternative algebras over an arbitrary field $\mathbb{F}$. We suggest a new method based on the sequence of differences between the dimensions of the linear spans of words,…
We consider the long-term dynamics of the vanishing stepsize subgradient method in the case when the objective function is neither smooth nor convex. We assume that this function is locally Lipschitz and path differentiable, i.e., admits a…
We introduce and study a new partial order on Dyck paths. We prove that these posets are meet-semilattices. We show that their numbers of intervals are the same as the number of bicubic planar maps. We describe an unexpected connection with…