Related papers: Lattice paths enumerations weighted by ascent leng…
Given two relatively prime positive integers $\alpha$ and $\beta$, we consider simple lattice paths (with unit East and unit North steps) from $(0,0)$ to $(\alpha k,\beta k)$, and enumerate them by their left and right bounces with respect…
We investigate the Whitney numbers of the first kind of rank-metric lattices, which are closely linked to the open problem of enumerating rank-metric codes having prescribed parameters. We apply methods from the theory of hyperovals and…
This paper studies three different ways to assign weights to the lattice points of a convex polytope and discusses the algebraic and combinatorial properties of the resulting weighted Ehrhart functions and their generating functions and…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of…
In this work, we define flattened Catalan words as Catalan words whose runs of weak ascents have leading terms that appear in weakly increasing order. We provide generating functions, formulas, and asymptotic expressions for the number of…
The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics…
The co-evolution between network structure and functional performance is a fundamental and challenging problem whose complexity emerges from the intrinsic interdependent nature of structure and function. Within this context, we investigate…
We analyze some enumerative and asymptotic properties of Dyck paths under a line of slope 2/5.This answers to Knuth's problem \\#4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June…
We characterize the inclusion relations between weighted classes of entire functions with rapid decreasing growth and study strong growth comparison relations between given weights. In our considerations first we focus on weights defined in…
We analyze the combinatorics behind the operation of taking the logarithm of the generating function $G_k$ for $k^\text{th}$ generalized Catalan numbers. We provide combinatorial interpretations in terms of lattice paths and in terms of…
It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fu{\ss}-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps.…
We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating…
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also…
Counting the number of permutations of a given total displacement is equivalent to counting weighted Motzkin paths of a given area (Guay-Paquet and Petersen, 2014). The former combinatorial problem is still open. In this work, we show that…
Studying the generalized Hamming weights of linear codes is a significant research area within coding theory, as it provides valuable structural information about the codes and plays a crucial role in determining their performance in…
We present a complete solution to the so-called tennis ball problem, which is equivalent to counting lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit…
Let $G$ be a directed graph on finitely many vertices and edges, and assign a positive weight to each edge on $G$. Fix vertices $u$ and $v$ and consider the set of paths that start at $u$ and end at $v$, self-intersecting in any number of…
Osculating paths are sets of directed lattice paths which are not allowed to cross each other or have common edges, but are allowed to have common vertices. In this work we derive a constant term formula for the number of such lattice paths…
We introduce a new approach to obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator being considered satisfies a certain type of weighted integral inequalities. The method is illustrated on…
Graphs have been commonly used to model many applications. A natural problem which abstracts applications such as itinerary planning, playlist recommendation, and flow analysis in information networks is that of finding the heaviest path(s)…