A bijection between $321$- and $213$-avoiding permutations preserving $t$-stack-sortability

TL;DR

This paper introduces a bijection between 321- and 213-avoiding permutations that preserves t-stack-sortability, providing a refined enumeration and connecting natural permutation statistics through binary trees.

Contribution

It constructs a novel bijection between two classes of pattern-avoiding permutations that maintains t-stack-sortability and related statistics, refining previous conjectures.

Findings

Bijection preserves t-stack-sortability.

Refines enumeration conjecture by Zhang and Kitaev.

Connects permutation statistics via binary trees.

Abstract

We construct a bijection between $321$- and $213$-avoiding permutations that preserves the property of $t$-stack-sortability. Our bijection transforms natural statistics between these two classes of permutations and proves a refinement of an enumerative conjecture posed by Zhang and Kitaev. This work contributes further to the long-standing line of research on bijections between length-3 pattern avoiding permutations. Increasing binary trees lie at the heart of our approach.