Lehmer Parking Functions and Their Outcomes

Abstract

We introduce Lehmer parking functions and study their set of parking outcomes. Our main results establish that the number of outcomes of Lehmer parking functions of length $n$ is given by a Bell number, which is exactly the number of set partitions of an $n$ element set. We also show that the number of outcomes of weakly decreasing Lehmer parking functions is given by a Catalan number, which corresponds to a subset of set partitions on a set with $n$ elements referred to as non-intersecting set partitions.

Figures (10)

• Figure 1: The grid diagram for the Lehmer parking function $\alpha=(5,2,4,2,1,1)$. The entry in each row lies at or below the antidiagonal given by $a_i=n-i+1$.
• Figure 2: Parking cars using the Lehmer parking function $\alpha=(5,2,4,2,1,1)$ results in the cars parking in order (i.e., in outcome) $524316\in\mathfrak{S}_6$. Car image designed by Freepik and callout designed by macrovector / Freepik.
• Figure 3: We park the preference list $\alpha=(2,2,1)$. Here, the dot $a_1=2$ on the diagram conveys that the preference of car $1$ is spot $2$. In the third diagram, car $2$ is forced to park in spot $3$ because spot $2$ is occupied by car $1$. The outcome of $\alpha$ is the permutation $\pi=312$.
• Figure 4: We display the desired outcome $\pi=132$ in blue. If car $1$ parks in spot $1$, it is impossible for car $2$ to park in spot $3$. Thus, $\pi$ is not the outcome of a Lehmer parking function.
• Figure 5: The arm-leg diagrams of $\pi=341526$ and $\rho=341625$, respectively. In blue we identify the entries $(i,\pi_i)$ or $(i,\rho_i)$. The right diagram is intersecting, which can be seen by its containment of the pattern $n-j+1<n-i+1\le\rho_j<\rho_i$ for $i=4$, $j=6$.

Theorems & Definitions (47)

• Definition 1.1
• Lemma 1.1
• proof
• Remark
• Definition 2.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.1