# A decorated tree approach to random permutations in substitution-closed classes

## TL;DR

This paper introduces a new bijective encoding of permutations as decorated trees, enabling analysis of their local and global limits within substitution-closed classes, and strengthening existing permuton limit results.

## Contribution

It presents a novel bijective encoding of permutations as decorated trees and applies it to prove local convergence and enhance permuton limit results for substitution-closed classes.

## Key findings

- Established local convergence of uniform random permutations in certain classes
- Reproved and strengthened permuton limit results using new methods
- Connected permutation limits with size-constrained Galton-Watson trees

## Abstract

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a criticality constraint. It also enables us to reprove and strengthen permuton limits for these classes in a new way, that uses a semi-local version of Aldous' skeleton decomposition for size-constrained Galton--Watson trees.