A note on a bijection for Schr\"oder permutations

TL;DR

This paper provides a concise description of a known bijection between Schr"oder paths and certain pattern-avoiding permutations, linking geometric and permutation statistics.

Contribution

It offers a simplified and clearer explanation of an existing bijection connecting Schr"oder paths with 4132, 4231-avoiding permutations.

Findings

Bijection maps area to inversion number

Simplified description of the bijection

Clarifies the relationship between paths and permutations

Abstract

There is a bijection from Schroder paths to {4132, 4231}-avoiding permutations due to Bandlow, Egge, and Killpatrick that sends "area" to "inversion number". Here we give a concise description of this bijection.