Related papers: The ascent lattice on Dyck paths
Lattice paths effectively model phenomena in chemistry, physics and probability theory. Asymptotic enumeration of lattice paths is linked with entropy in the physical systems being modeled. Lattice paths restricted to different regions of…
Returning walks on a lattice are sequences of moves that start at a given lattice site and return to the same site after $n$ steps. Determining the total number of returning walks of a given length $n$ is a typical graph-theoretical problem…
A quantum walk whose continuous limit coincides with Dirac equation is usually called a Dirac Quantum Walk (DQW). A new systematic method to build DQWs coupled to electromagnetic (EM) fields is introduced and put to test on several examples…
We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey \emph{two-step rules,} which allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a…
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…
Let $W_d(n)$ be the number of $2n$-step walks in $\mathbb{Z}^d$ which begin and end at the origin. We study the exponent of $2$ in the prime factorisation of this number; i.e., $w_d(n) = \nu_2(W_d(n))$. We show that, for each $d$, there is…
We continue the investigations of lattice walks in the three dimensional lattice restricted to the positive octant. We separate models which clearly have a D-finite generating function from models for which there is no reason to expect that…
Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works…
We study the $SU(\infty)$ lattice Yang-Mills theory at the dimensions $D=2,3,4$ via the numerical bootstrap method. It combines the Makeenko-Migdal loop equations, with a cut-off $L_{\mathrm{max}}$ on the maximal length of loops, and…
Let $L$ be a lattice of finite length and let $d$ denote the minimum path length metric on the covering graph of $L$. For any $\xi=(x_1,\dots,x_k)\in L^k$, an element $y$ belonging to $L$ is called a median of $\xi$ if the sum…
In a capacitated directed graph, it is known that the set of all min-cuts forms a distributive lattice [1], [2]. Here, we describe this lattice as a regular predicate whose forbidden elements can be advanced in constant parallel time after…
Generalized Dyck paths (or discrete excursions) are one-dimensional paths that take their steps in a given finite set S, start and end at height 0, and remain at a non-negative height. Bousquet-M\'elou showed that the generating function…
Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in $\{0, \pm 1\}^2$: the generating function is D-finite if…
This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. At each step, a nondeterministic walk draws a random set of steps from a predefined set of sets and explores all possible extensions in parallel.…
In the past decade, a lot of attention has been devoted to the enumera-tion of walks with prescribed steps confined to a convex cone. In two dimensions, this means counting walks in the first quadrant of the plane (possibly after a linear…
In this work, we introduce the new class of functions which can use to solve the nonlinear/linear multi-dimensional differential equations. Based on these functions, a numerical method is provided which is called the Developed Lagrange…
Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural…
Non-negative {\L}ukasiewicz paths are special two-dimensional lattice paths never passing below their starting altitude which have only one single special type of down step. They are well-known and -studied combinatorial objects, in…
Kim and Drake used generating functions to prove that the number of 2-distant noncrossing matchings, which are in bijection with little Schroeder paths, is the same as the weight of Dyck paths in which downsteps from even height have weight…
This work investigates the factorization of finite lattices to implode selected intervals while preserving the remaining order structure. We examine how complete congruence relations and complete tolerance relations can be utilized for this…