Related papers: Parking functions, multi-shuffle, and asymptotic p…

Let $W$ be a Weyl group with root lattice $Q$ and Coxeter number $h$. The elements of the finite torus $Q/(h+1)Q$ are called the $W$-{\sf parking functions}, and we call the permutation representation of $W$ on the set of $W$-parking…

Given an m-dimensional compact submanifold $\mathbf{M}$ of Euclidean space $\mathbf{R}^s$, the concept of mean location of a distribution, related to mean or expected vector, is generalized to more general $\mathbf{R}^s$-valued functionals…

We derive asymptotic expansions of the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed. For both functions we consider $b/a\le 1$ and $b/a\ge 1$, with special attention for the case…

The paper is concerned with asymptotic properties of the principal components analysis of functional data. The currently available results assume the existence of the fourth moment. We develop analogous results in a setting which does not…

An asymmetric exclusion process type process, where cars move forward along a closed road that starts and terminates at a parking garage, displays dynamic phase transitions into two types of condensate phases where the garage becomes…

In a recent paper J. Haglund showed that a certain symmetric function expresion enumerates by t^{area} q^{dinv} of the parking functions whose diagonal word is in the shuffle of 12...j and j+1...j+n with k of the cars j+1,...,j+n in the…

A classifier for two or more samples is proposed when the data are high-dimensional and the underlying distributions may be non-normal. The classifier is constructed as a linear combination of two easily computable and interpretable…

Let $M$ be a $d\times d$ contracting matrix. In this paper we consider the self-affine iterated function system $\{Mv-u, Mv+u\}$, where $u$ is a cyclic vector. Our main result is as follows: if $|\det M|\ge 2^{-1/d}$, then the attractor…

We study the probability that one beta-distributed random variable exceeds the maximum of two others, allowing all three to have general parameters. This amounts to studying Euler transforms of products of two incomplete beta functions. We…

Central configurations are solutions of the equations $\lambda m_j\boldsymbol{q}_j = \frac{\partial U}{\partial \boldsymbol{q}_j}$, where $U$ denotes the potential function and each $\boldsymbol{q}_j$ is a point in the $d$-dimensional…

We answer an open problem of arXiv:1204.1760 and arXiv:1205.4293, extending their work to irreducible well--generated complex reflection groups $W$. We define a combinatorial $W$-noncrossing parking space and an algebraic $W$-parking space…

Consider a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number $m \le n$ of cars arrive one by…

We investigate dissipative dynamical systems under the influence of an external driving with two or more frequencies. Our main quantities of interest are long-time averages of expectation values which turn out to exhibit universal features.…

The main purpose of this paper is to provide threshold functions for the events that a random subset of the points of a finite vector space has certain properties related to point-flat incidences. Specifically, we consider the events that…

Multiplicative invariance is a well-studied property of subsets of the unit interval. The theory in the complex plane is less developed. This paper introduces an analogous definition for multiplicative invariance in the complex plane…

In 1966, Konheim and Weiss [33] introduced a now classical parking protocol. The deterministic process and its resultant objects, known as parking functions, have since become a favorite object of study in enumerative combinatorics. In our…

We extend the notion of parking function polytopes and study their geometric and combinatorial structure, including normal fans, face posets, and $h$-polynomials, as well as their connections to other classes of polytopes. To capture their…

This work builds on the notion of record of rooted trees. We provide an alternative definition of parking functions, derive from it a record-preserving bijection between rooted trees and parking functions, and establish a join…

We introduce Lehmer parking functions and study their set of parking outcomes. Our main results establish that the number of outcomes of Lehmer parking functions of length $n$ is given by a Bell number, which is exactly the number of set…

For the Schr\"odinger equation $-d^2 u/dx^2 + q(x)u = \lambda u$ on a finite $x$-interval, there is defined an "asymmetry function" $a(\lambda;q)$, which is entire of order $1/2$ and type $1$ in $\lambda$. Our main result identifies the…