A bijection between vertices of rank 1 in increasing trees and derangements

TL;DR

This paper presents a combinatorial proof establishing a bijection between certain vertices in increasing trees and derangements, providing a new perspective on their structural relationship.

Contribution

It introduces a novel bijection linking neighbors of leaves in increasing trees to derangements, offering a combinatorial proof of Bóna's recent result.

Findings

Established a bijection between neighbors of leaves in increasing trees and derangements

Provided a combinatorial proof for a known result by Bóna

Enhanced understanding of the structural correspondence between trees and permutations

Abstract

We give a combinatorial proof of a recent result of B\'ona by constructing a bijection from the set of all neighbors of leaves of increasing trees of size $n$ to the set of derangements of length $n$.