A refined bijection between alternating permutations and 0-1-2 increasing trees

TL;DR

This paper introduces a precise bijection linking alternating permutations with 0-1-2 increasing trees, preserving the first element in a specific traversal order, thus deepening combinatorial understanding.

Contribution

It constructs a refined bijection between alternating permutations and 0-1-2 increasing trees that maintains the initial element correspondence.

Findings

Establishes a bijection preserving the first element in postorder traversal.

Provides a combinatorial framework connecting permutations and trees.

Enhances understanding of the structure of alternating permutations.

Abstract

We construct a refined bijection $\phi$ between alternating permutations and 0-1-2 increasing trees with degree at most 2. It satisfies that the first element of alternating permutation $\pi$ is equal to the first vertex in $\phi(\pi)$ in the postorder.