Related papers: Lattice paths with a first return decomposition co…

We recall the main types of lattice paths, which are sequences in the lattice of integer coordinates points in the plane. We start with the fundamental central lattice paths and Dyck paths and proceed in elementary terms through recently…

$k$-Dyck paths differ from ordinary Dyck paths by using an up-step of length $k$. We analyze at which level the path is after the $s$-th up-step and before the $(s+1)$st up-step. In honour of Rainer Kemp who studied a related concept 40…

We derive lower and upper bounds on possible growth rates of certain sets of positive integers $A_k=\{1= a_1 < a_2 < ... < a_{k}\}$ such that all integers $n\in \{0, 1, 2, ..., ka_{k}\}$ can be represented as a sum of no more than $k$…

Let $\mathcal{L}_n$ denote the set of all paths from $[0,0]$ to $[n, n]$ which consist of either unit north steps $N$ or unit east steps $E$ or, equivalently, the set of all words $L \in \{E,N\}^*$ with $n$ $E$'s and $n$ $N$'s. Given $L \in…

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…

We introduce a path-theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher level generalisations over fields of arbitrary characteristic. Our first main result is a…

A lattice path inside the $m\times n$ table $T$ is a sequence $\nu_1,\ldots,\nu_k$ of cells such that $\nu_{j+1}-\nu_j\in\{(1,-1),(1,0),(1,1)\}$ for all $j=1,\ldots,k-1$. The number of lattice paths in $T$ from the first column to the…

We define a map from subspaces to Motzkin paths and show that the inverse image of every path is a disjoint union of symmetric Boolean subsets yielding an explicit symmetric Boolean decomposition of the subspace lattice.

A variation of Dyck paths allows for down-steps of arbitrary length, not just one. This is motivated by ideas due to Emeric Deutsch. We use the adding-a-new-slice technique and the kernel method to compute the number of maximal runs of…

In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape.…

Various lattice path models are reviewed. The enumeration is done using generating functions. A few bijective considerations are woven in as well. The kernel method is often used. Computer algebra was an essential tool. Some results are…

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 initiate the study of a type $C_n$ generalization of the lattice path matroids defined by Bonin, de Mier, and Noy. These are delta matroids whose feasible sets are in bijection with lattice paths which are symmetric along the main…

In this paper, we study nonconvex constrained optimization problems with both equality and inequality constraints, covering deterministic and stochastic settings. We propose a novel first-order algorithm framework that employs a…

A \emph{Dyck path} is a lattice path in the first quadrant of the $xy$-plane that starts at the origin, ends on the $x$-axis, and consists of the same number of North-East steps $U$ and South-East steps $D$. A \emph{valley} is a subpath of…

Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…

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…

This article examines matrices whose entries are determined by recursive relations of the form $A_{i, j} = x A_{i, j-1} + y A_{i-1, j-1} + z A_{i-1, j}$, where $x, y, z$ are constants, and the initial conditions are defined along the first…

This short note deals with the so-called $ Sock \; Matching \; Problem$. We define $B_{n,k}$ as the number of all the finite sequences $a_1, \ldots, a_{2n}$ of nonnegative integers which contain at least one occurrence of $k$ $(1 \leq k…

Let $\mathcal{T}$ be a collection of 3-element subsets $S$ of $\{1, \ldots,n\}$ with the property that if $i<j<k$ and $a<b<c$ are two 3-element subsets in $S$, then there exists an integer sequence $x_1 < x_2 < \cdots < x_n$ such that $x_i,…