On Parking Functions and The Tower of Hanoi

TL;DR

This paper establishes a bijection between parking functions with displacement one and Tower of Hanoi states, revealing a surprising combinatorial connection and enumeration equivalence via Lah numbers.

Contribution

It introduces a novel bijection linking parking functions with displacement one to Tower of Hanoi configurations, expanding understanding of their combinatorial structures.

Findings

Parking functions with displacement one are enumerated by Lah numbers.

A bijection between these parking functions and Tower of Hanoi states is constructed.

The set of ideal states in Tower of Hanoi corresponds to a specific subset of parking functions.

Abstract

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 bijection with the set of ideal states in the famous Tower of Hanoi game with $n+1$ disks and $n+1$ pegs, both sets being enumerated by the Lah numbers.