# Roller Coaster Permutations and Partition Numbers

## TL;DR

This paper investigates the partition properties of roller coaster permutations, establishing an upper bound on their partition numbers and providing experimental data that suggests the bound is nearly optimal.

## Contribution

It introduces a theoretical upper bound for the partition number of roller coaster permutations and connects their structure to these bounds.

## Key findings

- Derived an upper bound for partition numbers of roller coaster permutations.
- Experimental data for small n supports the near-sharpness of the bound.
- Connected permutation structure to partition number properties.

## Abstract

This paper explores the partition properties of roller coaster permutations, a class of permutations characterized by maximizing the number of alternating runs in all subsequences. We establish a connection between the structure of these permutations and their partition numbers, defined as the minimum number of monotonic subsequences required to cover the permutation. Our main result provides a theoretical upper bound for the partition number of a roller coaster permutation of length $n$, given by $P_{max}(n) \le \lfloor\frac{\lceil\frac{n-2}{2}\rceil}{2}\rfloor + 2$. We further present experimental data for $n < 15$ that suggests this bound is nearly sharp.