Lehmer Parking Functions and Their Outcomes

TL;DR

This paper introduces Lehmer parking functions, revealing that their outcomes are counted by Bell numbers, and explores a special case where outcomes relate to Catalan numbers, connecting parking functions to classical combinatorial structures.

Contribution

The paper defines Lehmer parking functions and establishes their outcome counts as Bell numbers and Catalan numbers for specific subclasses, linking parking functions to set partitions.

Findings

Number of outcomes of Lehmer parking functions of length n equals the Bell number.

Outcomes of weakly decreasing Lehmer parking functions are counted by Catalan numbers.

Connects parking functions to set partitions and non-intersecting partitions.

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.