On a Family of Conjectures of Joel Lewis on Alternating Permutations

TL;DR

This paper proves generalized conjectures about the count of alternating permutations avoiding specific patterns, revealing that a classic bijection often preserves the alternating property, thus advancing understanding in permutation pattern avoidance.

Contribution

It introduces generalized versions of Joel Lewis's conjectures and shows that a classic bijection preserves the alternating property in pattern-avoiding permutations.

Findings

Generalized conjectures on alternating permutations were proven.

A classic bijection preserves the alternating property.

Results extend understanding of pattern avoidance in permutations.

Abstract

We prove generalized versions of some conjectures of Joel Lewis on the number of alternating permutations avoiding certain patterns. Our main tool is the perhaps surprising observation that a classic bijection on pattern avoiding permutations often preserves the alternating property.