Related papers: Parking Functions on Directed Graphs and Some Dire…
We define a "shifted analogue" $\mathrm{SH}_n$ of the parking function symmetric function $\mathrm{PF}_n$. The expansion of $\mathrm{SH}_n$ in terms of three bases for shifted symmetric functions is explicitly described. We don't know a…
A parking function of length $n$ is a sequence $\pi=(\pi_1,\dots, \pi_n)$ of positive integers such that if $\lambda_1\leq\cdots\leq \lambda_n$ is the increasing rearrangement of $\pi_1,\dots,\pi_n$, then $\lambda_i\leq i$ for $1\leq i\leq…
Let $1\leq r\leq n$ and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labelled tree on $n+1$ vertices, exactly $r$ vertices are visited before backtracking. Let $R$ be the set of trees with this property.…
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…
Naples parking functions were introduced as a generalization of classical parking functions, in which cars are allowed to park backwards, by checking up to a fixed number of previous slots, before proceedings forward as usual. In our…
The conceptions of $G$-parking functions and $G$-multiparking functions were introduced in [15] and [12] respectively. In this paper, let $G$ be a connected graph with vertex set $\{1,2,...,n\}$ and $m\in V(G)$. We give the definition of…
Transit functions were introduced as models of betweenness on undirected structures. Here we introduce directed transit function as the directed analogue on directed structures such as posets and directed graphs. We first show that…
For a directed graph G on vertices {0,1,...,n}, a G-parking function is an n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty subset U of {1,...,n}, there exists a vertex j in U for which there are more than b_j…
There are several combinatorial objects that are known to be in bijection to the spanning trees of a graph G. These objects include G-parking functions, critical configurations of G, and descending traversals of G. In this paper, we extend…
In this paper, we view parking functions viewed as labeled Dyck paths in order to study a notion of pattern avoidance first introduced by Remmel and Qiu. In particular we enumerate the parking functions avoiding any set of two or more…
A parking function of length n is a sequence (b_1, b_2,..., b_n) of nonnegative integers whose nondecreasing rearrangement (a_1, a_2,...,a_n) has the property that a_i < i for every i. A well-known result about parking functions is that the…
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 extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences.
In this paper, let $\mathcal{P}_{n,n+k;\leq n+k}$ (resp. $\mathcal{P}_{n;\leq s}$) denote the set of parking functions $\alpha=(a_1,...,a_n)$ of length $n$ with $n+k$ (respe. $n$)parking spaces satisfying $1\leq a_i\leq n+k$ (resp. $1\leq…
Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot…
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 introduce a generalization of parking functions in which cars are limited in their movement backwards and forwards by two nonnegative integer parameters $k$ and $\ell$, respectively. In this setting, there are $n$ spots on a one-way…
Interval parking functions (IPFs) are a generalization of ordinary parking functions in which each car is willing to park only in a fixed interval of spaces. Each interval parking function can be expressed as a pair $(a,b)$, where $a$ is a…
Consider a rooted tree on the top of which we let cars arrive on its vertices. Each car tries to park on its arriving vertex but if it is already occupied, it drives towards the root of the tree and parks as soon as possible. In this…
A parking function of length $n$ is prime if we obtain a parking function of length $n-1$ by deleting one 1 from it. In this note we give a new direct proof that the number of prime parking functions of length $n$ is $(n-1)^{n-1}$. This…