Related papers: On some discrete statistics of parking functions
We recall that the $k$-Naples parking functions of length $n$ (a generalization of parking functions) are defined by requiring that a car which finds its preferred spot occupied must first back up a spot at a time (up to $k$ spots) before…
The displacement of a car with respect to a parking function is the number of spots it must drive past its preferred spot in order to park. An $\ell$-interval parking function is one in which each car has displacement at most $\ell$. Among…
In this paper we study the asymptotic behavior of a random uniform parking function $\pi_n$ of size $n$. We show that the first $k_n$ places $\pi_n(1),\dots,\pi_n(k_n)$ of $\pi_n$ are asymptotically i.i.d. and uniform on $\{1,2,\dots,n\}$,…
We study the asymptotic behavior of cycles of uniformly random parking functions. Our results are multifold: we obtain an explicit formula for the number of parking functions with a prescribed number of cyclic points and show that the…
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…
We enumerate interlaced pairs of parking functions whose underlying Dyck path has a bounded height. We obtain an explicit formula for this enumeration in the form of a quotient of analogs of Chebicheff polynomials having coefficients in the…
The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length $n$ with displacement one is in…
Parking sequences (a generalization of parking functions) are defined by specifying car lengths and requiring that a car attempts to park in the first available spot after its preference. If it does not fit there, then a collision occurs…
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…
We give a very short proof of the fact that the number of $(a,b)$-parking functions of length $n$ equals $a(a+bn)^{n-1}$. This was first proved in 2003 by Kung and Yan, via a very long and torturous route, as a corollary of a more general…
Fubini rankings with $n$ competitors are $n$-tuples with entries in $[n]=\{1,2,3,\ldots, n\}$ that encode the conclusion of a race that allows ties. Since Fubini rankings are parking functions, we can study their parking outcomes, which are…
A parking function on $[n]$ creates a permutation in $S_n$ via the order in which the $n$ cars appear in the $n$ parking spaces. Placing the uniform probability measure on the set of parking functions on $[n]$ induces a probability measure…
Let $\mathfrak{S}_n$ denote the symmetric group and let $W(\mathfrak{S}_n)$ denote the weak order of $\mathfrak{S}_n$. Through a surprising connection to a subset of parking functions, which we call unit Fubini rankings, we provide a…
In a parking function, a car is considered lucky if it is able to park in its preferred spot. Extending work of Harris and Martinez, we enumerate outcomes of parking functions with a fixed set of lucky cars. We then consider a…
We propose a characterization of $k$-Naples parking functions in terms of subsequences with the structure of a complete $k$-Naples parking function. We define complete parking preferences by requiring that for all $j=2,\dots,n$, the number…
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 $[n]$.
An $(m, n)$-parking function can be characterized as function $f:[n] \to [m]$ such that the partition obtained by reordering the values of $f$ fits inside a right triangle with legs of length $m$ and $n$. Recent work by McCammond, Thomas,…
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…
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 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.