相关论文: Parking functions on directed stars and orientatio…
Recall that $\alpha=(a_1,a_2,\ldots,a_n)\in[n]^n$ is a parking function if its nondecreasing rearrangement $\beta=(b_1,b_2,\ldots,b_n)$ satisfies $b_i\leq i$ for all $1\leq i\leq n$. In this article, we study parking functions based on…
Using measured data we demonstrate that there is an amazing correspondence among the statistical properties of spacings between parked cars and the distances between birds perching on a power line. We show that this observation is easily…
We generalize the concept of ascending and descending runs from permutations to rooted labelled trees and mappings, i.e., functions from the set $\{1, \dots, n\}$ into itself. A combinatorial decomposition of the corresponding functional…
In genome rearrangement theory, one of the elusive questions raised in recent years is the enumeration of rearrangement scenarios between two genomes. This problem is related to the uniform generation of rearrangement scenarios, and the…
We show that the number of parking functions of length $n$ with zero secondary dinv is equal to the number of ordered cycle decompositions of permutations of $[n]$.
To any real rational function with generic ramification points we assign a combinatorial object, called a garden, which consists of a weighted labeled directed planar chord diagram and of a set of weighted rooted trees each corresponding to…
We enumerate a class of fully parked trees. In a probabilistic context, this means computing the partition function $F(x,y)$ of the parking process where an i.i.d. number of cars arrives at each vertex of a Galton-Watson tree with a…
Rapidly-exploring random tree (RRT) has been applied for autonomous parking due to quickly solving high-dimensional motion planning and easily reflecting constraints. However, planning time increases by the low probability of extending…
In this article, we establish new results on the probabilistic parking model (introduced by Durm\'ic, Han, Harris, Ribeiro, and Yin) with $m$ cars and $n$ parking spots and probability parameter $p\in[0,1]$. For any $ m \leq n$ and $p \in…
Due to possibly changing pose of a movable object and nonholonomic constraint of a differential-drive robot, it is challenging to design an object servoing scheme for the differential-drive robot to asymptotically park at a predefined…
We show that the notion of parkization of a word, a variant of the classical standardization, allows to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and…
It is known that there exists a function interpolating a given data set such that the graph of the function is the attractor of an iterated function system which is called fractal interpolation function. We generalize the notion of fractal…
Let $\mathcal{P}_n$ be the convex hull in $\mathbb{R}^n$ of all parking functions of length $n$. Stanley found the number of vertices and the number of facets of $\mathcal{P}_n$. Building upon these results, we determine the number of faces…
This work investigates the duality between two discrete dynamical processes: parking functions, and the Abelian sandpile model (ASM). Specifically, we are interested in the extension of classical parking functions, called $G$-parking…
Real-time monitoring of traffic density, road congestion, public transportation, and parking availability are key to realizing the vision of a smarter city and, with the advent of vehicular networking technologies such as IEEE 802.11p and…
Initially a car is placed with probability p at each site of the two-dimensional integer lattice. Each car is equally likely to be East-facing or North-facing, and different sites receive independent assignments. At odd time steps, each…
Stanley recently introduced the shifted parking function symmetric function $SH_n$, which is the shiftification of Haiman's parking function symmetric function $PF_n$. The function $SH_n$ lives in the subalgebra of symmetric functions…
Orienting surface normals correctly and consistently is a fundamental problem in geometry processing. Applications such as visualization, feature detection, and geometry reconstruction often rely on the availability of correctly oriented…
We focus on Optimal Transport PDE on the unit sphere $\mathbb{S}^2$ with a particular type of cost function $c(x,y) = F(x \cdot y, x \cdot \hat{e}, y \cdot \hat{e})$ which we call cost functions with preferential direction, where $\hat{e}…
We introduce several associative algebras and series of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we a series of representations…