Related papers: Uniform asymptotics of area-weighted Dyck paths

We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of…

We consider a generalised version of Motzkin paths, where horizontal steps have length $\ell$, with $\ell$ being a fixed positive integer. We first give the general functional equation for the area-length generating function of this model.…

Randomly sampling an acyclic orientation on the complete bipartite graph $K_{n,k}$ with parts of size $n$ and $k$, we investigate the length of the longest path. We provide a probability generating function for the distribution of the…

Consider a non-negative sequence $c_n = h(n) \cdot n^{\alpha-1} \cdot \rho^{-n}$, where $h$ is slowly varying, $\alpha>0$, $0<\rho<1$ and $n\in\mathbb{N}$. We investigate the coefficients of $G(x,y) = \prod_{k\ge1}(1-x^ky)^{-c_k}$, which is…

We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, 'jumps' orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with…

We obtain bivariate asymptotics for the number of (unicellular) combinatorial maps (a model of discrete surfaces) as both the size and the genus grow. This work is related to two research topics that have been very active recently:…

In this paper, we first describe how to find the generating function for the sum of the areas under generalized Dyck paths (with an arbitrary set of steps) using Motzkin paths as a motivating example. We then focus on Motzkin and Dyck…

This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics…

We derive asymptotic formulae for the coefficients of bivariate generating functions with algebraic and logarithmic factors. Logarithms appear when encoding cycles of combinatorial objects, and also implicitly when objects can be broken…

A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's…

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic…

Area-weighted Dyck-paths are a two-dimensional model for vesicles attached to a wall. We model the mechanical response of a vesicle to a pulling force by extending this model. We obtain an exact solution using two different approaches,…

We derive the length and area generating function of planar height-restricted forward-moving discrete paths of increments +1, 0, or -1 with arbitrary starting and ending points, the so-called Motzkin meanders, and the more general…

Decay to asymptotic steady state in one-dimensional logistic-like mappings is characterized by considering a phenomenological description supported by numerical simulations and confirmed by a theoretical description. As the control…

In a Dyck path a peak which is (weakly) higher than all the preceding peaks is called a strict (weak) left to right maximum. We obtain explicit generating functions for both weak and strict left to right maxima in Dyck paths. The proofs of…

We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[{}_2F_1(a+\epsilon\lambda,b;c+\lambda;x),\qquad 0<x<1\] as $\lambda\to+\infty$ in the neigbourhood of $\epsilon x=1$ when the parameter $\epsilon>1$ and…

We consider a bivariate rational generating function F(x,y) = P(x,y) / Q(x,y) = sum_{r, s} a_{r,s} x^r y^s under the assumption that the complex algebraic curve $\sing$ on which $Q$ vanishes is smooth. Formulae for the asymptotics of the…

We analyse weighted Motzkin paths with step multiplicities that vary linearly with height. In the balanced case the associated exponential generating function satisfies a Pearson-type PDE, and solving by characteristics yields closed…

Let $F(x)= \sum_{\nu\in\NN^d} F_\nu x^\nu$ be a multivariate power series with complex coefficients that converges in a neighborhood of the origin. Assume $F=G/H$ for some functions $G$ and $H$ holomorphic in a neighborhood of the origin.…

Let $\sum_{\beta\in\nats^d} F_\beta x^\beta$ be a multivariate power series. For example $\sum F_\beta x^\beta$ could be a generating function for a combinatorial class. Assume that in a neighbourhood of the origin this series represents a…