Note on permutations and permutation trees

Liu, Katherine; Wang, Hua

doi:10.2140/involve.2026.19.709

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Note on permutations and permutation trees

Katherine Liu and Hua Wang

Vol. 19 (2026), No. 4, 709–719

DOI: 10.2140/involve.2026.19.709

Abstract

Permutations have been studied in many contexts, and there are even multiple definitions of the term “permutation trees”. We use the definition first created by Luschny (2010) and recently considered by Elder et al. (2024), which develops a bijection between any permutation of $[n]$ and a labeled rooted tree with root 0 and $n$ vertices, and study the properties of these trees. Among our observations, we establish a novel conjugation between permutation trees which induce a reverse of its associated permutation and further expand this property by considering self-conjugation with respect to shape. In addition, we briefly discuss the behaviors of permutation trees corresponding to specific features of permutations and specific permutations such as payphone permutations.

Keywords

permutations, permutation trees

Mathematical Subject Classification

Primary: 05A05

Milestones

Received: 12 November 2024

Accepted: 22 March 2025

Published: 13 October 2026

Communicated by Jim Haglund

Authors

Katherine Liu
Massachusetts Institute of Technology Cambridge, MA United States

Hua Wang
Department of Mathematical Sciences Georgia Southern University Statesboro, GA United States

Vol. 19, No. 4, 2026