A Note on the Structure of Roller Coaster Permutations

TL;DR

This paper explores the structure of roller coaster permutations, focusing on their alternating and recursively alternating properties, and analyzes the distribution of entries in different positions within these permutations.

Contribution

It introduces the concept of recursively alternating permutations, a stronger condition than alternating, and examines entry behaviors in various positions within roller coaster permutations.

Findings

Recursively alternating permutations are a stricter subset of roller coaster permutations.

Entries in even, odd, and end positions exhibit specific structural behaviors.

The alternating structure influences the permutation's maximum switch properties.

Abstract

We consider the structure of roller coaster permutations as introduced by Ahmed & Snevily[1]. A roller coaster permutation is described as a permuta- tion that maximizes the total switches from ascending to descending or visa versa for the permutation and all of its subpermutations simultaneously. This paper looks at the alternating structure of these permutations and then we introduce a notion of a condition stronger than alternating for a permutation that we shall refer to as recursively alternating. We also examine the behav- ior of what entries can show up in even, odd, and end positions within the permutations.