Two permutation classes enumerated by the central binomial coefficients

TL;DR

This paper establishes a bijection between two specific permutation classes avoiding certain patterns and Dyck prefixes, enabling detailed analysis of permutation statistics and their enumeration by central binomial coefficients.

Contribution

It introduces a novel bijection linking pattern-avoiding permutations to Dyck prefixes, facilitating enumeration and statistical analysis of these classes.

Findings

Permutation classes are enumerated by central binomial coefficients.

The distribution of key permutation statistics is characterized.

A new combinatorial bijection is constructed.

Abstract

We define a map between the set of permutations that avoid either the four patterns $3214,3241,4213,4231$ or $3124,3142,4123,4132$, and the set of Dyck prefixes. This map, when restricted to either of the two classes, turns out to be a bijection that allows us to determine some notable features of these permutations, such as the distribution of the statistics "number of ascents", "number of left-to-right maxima", "first element", and "position of the maximum element"