Related papers: Parking functions, multi-shuffle, and asymptotic p…
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…
The notion of parking sequences is a new generalization of parking functions introduced by Ehrenborg and Happ. In the parking process defining the classical parking functions, instead of each car only taking one parking space, we allow the…
We recall that a parking function of length $n+1$ is said to be prime if removing any instance of 1 yields a parking function of length $n$. In this article, we study prime parking functions from multiple lenses. We derive an explicit…
For $\mathbf{b}=(b_1,\dots,b_n)\in \mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is defined to be a sequence $(\beta_1,\dots,\beta_n)$ of positive integers whose nondecreasing rearrangement $\beta'_1\leq \beta'_2\leq \cdots \leq…
A classical parking function of length $n$ is a list of positive integers $(a_1, a_2, \ldots, a_n)$ whose nondecreasing rearrangement $b_1 \leq b_2 \leq \cdots \leq b_n$ satisfies $b_i \leq i$. The convex hull of all parking functions of…
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…
In a parking function, a lucky car is a car that parks in its preferred parking spot and the parking outcome is the permutation encoding the order in which the cars park on the street. We give a characterization for the set of parking…
We consider the inversion enumerator I_n(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = -1 into this generalized polynomial…
Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call…
We propose a generalized car parking problem where either a car of size $\sigma$ or of size $m\sigma$ ($m>1$) is sequentially parked on a line with probability $q$ and $(1-q)$, respectively. The free parameter $q$ interpolates between the…
Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has…
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…
Parking functions are tuples that describe the parking of $M$ cars on a street with $M$ parking spots. In this paper, we define exact $k$-typed parking functions ($k$-TPFs) to be a variant of classical parking functions. We then establish…
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…
Kreweras proved that the reversed sum enumerator for parking functions of length $n$ is equal to the inversion enumerator for labeled trees on $n+1$ vertices. Recently, Perkinson, Yang, and Yu gave a bijective proof of this equality that…
In 2000, it was demonstrated that the set of $x$-parking functions of length $n$, where $x$=($a,b,...,b$) $\in \mathbbm{N}^n$, is equivalent to the set of rooted multicolored forests on [$n$]=\{1,...,$n$\}. In 2020, Yue Cai and Catherine H.…
In this paper, we obtain a q-exponential generating function for inversions on parking functions via symmetric function theory and also through a direct bijection to rooted labeled forests. We then apply these techniques to unit interval…
Parking functions, classically defined in terms of cars with preferred parking spots on a directed path attempting to park there, arise in many combinatorial situations and have seen various generalizations. In particular, parking functions…
Given a strictly increasing sequence $\mathbf{t}$ with entries from $[n]:=\{1,\ldots,n\}$, a parking completion is a sequence $\mathbf{c}$ with $|\mathbf{t}|+|\mathbf{c}|=n$ and $|\{t\in \mathbf{t}\mid t\le i\}|+|\{c\in \mathbf{c}\mid c\le…
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 spots, before proceeding forward as usual. In this work…