Related papers: Lehmer Parking Functions and Their Outcomes

It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain…

Given a positive-integer-valued vector $u=(u_1, \dots, u_m)$ with $u_1<\cdots<u_m$. A $u$-parking function of length $m$ is a sequence $\pi=(\pi_1, \dots, \pi_m)$ of positive integers whose non-decreasing rearrangement $(\lambda_1, \dots,…

Classical parking functions are defined as the parking preferences for $n$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $1$ to $n$ (from west to east). Cars drive down the street toward their…

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…

In the classical parking problem, unit intervals ("car lengths") are placed uniformly at random without overlapping. The process terminates at saturation, i.e. until no more unit intervals can be stowed. In this paper, we present a…

Lehmer's totient problem consists of determining the set of positive integers $n$ such that $\varphi(n)|n-1$ where $\varphi$ is Euler's totient function. In this paper we introduce the concept of $k$-Lehmer number. A $k$-Lehmer number is a…

In this expository article I describe classical results in the combinatorics of parking functions. Its English-Spanish translation is included. -- -- En este art\'iculo de difusi\'on matem\'atica describo resultados cl\'asicos en la…

We derive closed formulas for the number of $k$-coloured partitions and the number of plane partitions of $n$ in terms of the Bell polynomials.

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…

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…

In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the…

An ordered set-partition (or preferential arrangement) of n labeled elements represents a single ``hierarchy''; these are enumerated by the ordered Bell numbers. In this note we determine the number of ``hierarchical orderings'' or…

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…

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 study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions $S$ of subsets of $\mathbb{R}^d$ and of point sets that are (almost) subadditive in their first…

The Lempert function for a set of poles in a domain of $\mathbb C^n$ at a point $z$ is obtained by taking a certain infimum over all analytic disks going through the poles and the point $z$, and majorizes the corresponding multi-pole…

This paper provides an exploration of parking functions, a classical combinatorial object. We present two viewpoints on their structure and properties: through poset of noncrossing partitions and polytopes.

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…

Let $W$ be a Weyl group with root lattice $Q$ and Coxeter number $h$. The elements of the finite torus $Q/(h+1)Q$ are called the $W$-{\sf parking functions}, and we call the permutation representation of $W$ on the set of $W$-parking…

In this paper, we study classes of subexcedant functions enumerated by the Bell numbers and present bijections on set partitions. We present a set of permutations whose transposition arrays are the restricted growth functions, thus defining…