Related papers: Metered Parking Functions
Recent work on recurrence in quantum walks has provided a representation of Schur functions in terms of unitary operators. We propose a generalization of Schur functions by extending this operator representation to arbitrary operators on…
For any irreducible real reflection group $W$ with Coxeter number $h$, Armstrong, Reiner, and the author introduced a pair of $W \times \ZZ_h$-modules which deserve to be called {\sf $W$-parking spaces} which generalize the type A notion of…
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…
Intrinsic location functional is a large class of random locations containing locations that one may encounter in many cases, e.g., the location of the path supremum/infimum over a given interval, the first/last hitting time, etc. It has…
Learning embedding functions, which map semantically related inputs to nearby locations in a feature space supports a variety of classification and information retrieval tasks. In this work, we propose a novel, generalizable and fast method…
We introduce type C parking functions, encoded as vertically labelled lattice paths and endowed with a statistic dinv'. We define a bijection from type C parking functions to regions of the Shi arrangement of type C, encoded as diagonally…
Haglund, Morse, and Zabrocki introduced a family of creation operators of Hall-Littlewood polynomials, $\{C_{a}\}$ for any $a\in \mathbb{Z}$, in their compositional refinement of the shuffle (ex-)conjecture. For any $\alpha\vDash n$, the…
Tiered trees were introduced as a combinatorial object for counting absolutely indecomposable representation of certain quivers and torus orbit of certain homogeneous variety. In this paper, we define a bijection between the set of…
We study arithmetic functions $\Phi(x;d,a)$, called prime running functions, whose value at $x$ sums the gaps between primes $p_k \equiv a\ (\text{mod}\ d)$ below $x$ and the next following prime $p_{k+1}$, up to $x$. (The following prime…
Windowed recurrences are sliding window calculations where a function is applied iteratively across the window of data, and are ubiquitous throughout the natural, social, and computational sciences. In this monograph we explore the…
Parking occupancy in the area is defined by three major parameters - the rate of cars arrivals, the dwell time of already parked cars, and the willingness of drivers who are searching but yet did not find a vacant parking spot, to continue…
We introduce a basis of rational polynomial-like functions $P_0,\ldots,P_{n-1}$ for the free module of functions $Z/nZ\to Z/mZ$. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the…
In the Page parking (or packing) model on a discrete interval (also known as the discrete R{\'e}nyi packing problem or the unfriendly seating problem), cars of length two successively park uniformly at random on pairs of adjacent places,…
We study a class of stochastic processes of the type $\frac{d^n x}{dt^n}= v_0\, \sigma(t)$ where $n>0$ is a positive integer and $\sigma(t)=\pm 1$ represents an `active' telegraphic noise that flips from one state to the other with a…
A 0-1 matrix $M$ contains a 0-1 matrix pattern $P$ if we can obtain $P$ from $M$ by deleting rows and/or columns and turning arbitrary 1-entries into 0s. The saturation function $\mathrm{sat}(P,n)$ for a 0-1 matrix pattern $P$ indicates the…
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…
The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over $(m,n)$-parking functions are related to…
The action of the symmetric group $S_n$ on the set $Park_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $Park_n$ extends to an action of $S_{n+1}$.…
We study the parking process on the random recursive tree. We first prove that although the random recursive tree has a non-degenerate Benjamini--Schramm limit, the phase transition for the parking process appears at density $0$. We then…
In monitoring, we algorithmically check if a single behavior satisfies a property. Here, we consider monitoring for Multi-Lane Spatial Logic (MLSL). The behavior is given as a finite transition sequence of MLSL and the property is that a…