Related papers: Lattice paths with infinitely many down steps -- t…
We study lattice walks in a Weyl chamber of type A with fixed or free end points. For lattice walk models with zero drift that may be counted by means of a reflection argument, we determine asymptotics for the number of such walks as their…
Asinowski, Bacher, Banderier and Gittenberger (A. Asinowski, A. Bacher, C. Banderier and B. Gittenberger. Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown…
A rook path is a path on lattice points in the plane in which any proper horizontal step to the right or vertical step north is allowed. If, in addition, one allow bishop steps, that is, proper diagonal steps of slope 1, then one has queen…
Motzkin paths are simple yet important combinatorial objects. In this paper, we consider families of Motzkin paths with restrictions on peak heights, valley heights, upward-run lengths, downward-run lengths, and flat-run lengths. This paper…
We consider a random walk on a multidimensional integer lattice with random bounds on local times, conditioned on the event that it hits a high level before its death. We introduce an auxiliary "core" process that has a regenerative…
We discuss the structure of positive definite kernels in terms of operator models. In particular, we introduce two models, one of Hessenberg type and another one that we call near triangular. These models produce parametrizations of the…
Solutions to linear controlled differential equations can be expressed in terms of iterated path integrals of the driving path. This collection of iterated integrals encodes essentially all information about the driving path. While upper…
We consider the lattice of all the weak factorization systems on a given finite lattice. We prove that it is semidistributive, trim and congruence uniform. We deduce a graph theoretical approach to the problem of enumerating transfer…
We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine…
We introduce and study a deterministic lattice model describing the motion of an infinite system of oppositely charged particles under the action of a constant electric field. As an application this model represents a traffic flow of cars…
$k$-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…
In this paper we study variations of an old result by M\"{u}ller, Reiterman, and the last author stating that a countable graph has a subgraph with infinite degrees if and only if in any labeling of the vertices (or edges) of this graph by…
We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these…
Consider a randomly-oriented two dimensional Manhattan lattice where each horizontal line and each vertical line is assigned, once and for all, a random direction by flipping independent and identically distributed coins. A deterministic…
We study the number of 231-avoiding permutations with $j$-descents and maximum drop is less than or equal to $k$ which we denote by $a_{n,231,j}^{(k)}$. We show that $a_{n,231,j}^{(k)}$ also counts the number of Dyck paths of length $2n$…
Instead of $k$-Dyck paths we consider the equivalent concept of $k$-non-crossing trees. This is our preferred approach relative to down-step statistics modulo $k$ (first studied by Heuberger, Selkirk, and Wagner by different methods). One…
Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in…
We study a class of observables in four-dimensional superconformal Yang--Mills theories which, in the planar limit at finite 't Hooft coupling, can be expressed as determinants of semi-infinite matrices built from Bessel functions. This…
In this paper we studied infinite weighted automata and a general methodology to solve a wide variety of classical lattice path counting problems in an uniform way. This counting problems are related to Dyck paths, Motzkin paths and some…
Discrete differential equations appear most prominently in planar map and lattice path enumeration. In this work we consider discrete differential equations with an additional parameter $x$, where the order of the equation is $1$ for $x=0$…