Related papers: Lattice paths enumerations weighted by ascent leng…
The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths:…
In the field of enumeration of weighted walks confined to the quarter plane, it is known that the generating functions behave very differently depending on the chosen step set; in practice, the techniques used in the literature depend on…
We present a bijection between two well-known objects in the ubiquitous Catalan family: non-decreasing parking functions and {\L}ukasiewicz paths. This bijection maps the maximal displacement of a parking function to the height of the…
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…
The combinatorics of certain osculating lattice paths is studied, and a relationship with oscillating tableaux is obtained. More specifically, the paths being considered have fixed start and end points on respectively the lower and right…
We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given…
A parking function is a sequence $(a_1,\dots, a_n)$ of positive integers such that if $b_1\leq\cdots\leq b_n$ is the increasing rearrangement of $a_1,\dots,a_n$, then $b_i\leq i$ for $1\leq i\leq n$. In this paper we obtain some new results…
Lattice walks are used to model various physical phenomena. In particular, walks within Weyl chambers connect directly to representation theory via the Littelmann path model. We derive asymptotics for centrally weighted lattice walks within…
We extend the notion of nonbacktracking walks from unweighted graphs to graphs whose edges have a nonnegative weight. Here the weight associated with a walk is taken to be the product over the weights along the individual edges. We give two…
In this note we observe that a bijection related to Littelmann's root operators (for type $A_1$) transparently explains the well known enumeration by length of walks on $\N$ (left factors of Dyck paths), as well as some other enumerative…
We discuss the complexity of path enumeration and counting in weighted temporal graphs. In a weighted temporal graph, each edge has an availability time, a traversal time and some real cost. We introduce two bicriteria temporal min-cost…
We introduce a new type of lattice path, called brick-wall lattice path, and we derive a formula which counts the number of paths on these lattices imposing certain restrictions on the Cartesian plane. Connections to the Fibonacci sequence,…
In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevi\`ere, where cover relations…
We characterize the inclusions of weighted classes of entire functions in terms of the defining weights resp. weight systems. First we treat weights defined in terms of a so-called associated weight function where the weight(system) is…
We introduce the priority lattice, a structure arising from the priority search algorithm on rooted trees and forests. We prove bijectively that its maximal chains are labeled by parking functions, and that the maximal chains of its…
We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result extends well known enumerative formulae concerning lattice paths, and its derivation involves a classical…
We introduce weighted succession rules and parametric production matrices - simple extensions of the standard ECO method succession rules and production matrices. The purpose is to enumerate combinatorial objects with respect to several…
We show that the known matrix representations of the stationary state algebra of the Asymmetric Simple Exclusion Process (ASEP) can be interpreted combinatorially as various weighted lattice paths. This interpretation enables us to use the…
We show that weighted path orders are special instances of a variant of semantic path orders. Exploiting this fact, we introduce a generalization of weighted path orders that goes beyond the realm of simple termination. Experimental data…
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…