Bijections on rooted trees with fixed size of maximal decreasing subtrees

TL;DR

This paper provides a bijective proof connecting rooted trees with a fixed size of maximal decreasing subtrees to functions with specific image properties, enriching combinatorial understanding.

Contribution

It introduces a bijective proof for a known enumeration theorem relating rooted trees and functions, offering new combinatorial insights.

Findings

Established a bijection between rooted trees and functions with certain image constraints

Confirmed the enumeration formula for rooted trees with fixed maximal decreasing subtree size

Enhanced combinatorial understanding of tree-function correspondences

Abstract

Seo and Shin showed that the number of rooted trees on $[n+1]$ such that the maximal decreasing subtree with the same root has $k+1$ vertices is equal to the number of functions $f:[n]\to[n]$ such that the image of $f$ contains $[k]$. We give a bijective proof of this theorem.