Limiting distributions for the number of inversions in labelled tree families

TL;DR

This paper investigates the distribution of inversions in various families of labelled trees, providing both global and local asymptotic results for the total and node-specific inversions.

Contribution

It introduces a unified approach to analyze inversion distributions across multiple labelled tree families, including new limiting distribution results.

Findings

Limiting distributions for total inversions in labelled trees

Asymptotic behavior of node-specific inversions

Unified framework applicable to various tree families

Abstract

We consider so-called simple families of labelled trees, which contain, e.g., ordered, unordered, binary and cyclic labelled trees as special instances, and study the global and local behaviour of the number of inversions. In particular we obtain limiting distribution results for the total number of inversions as well as the number of inversions induced by the node labelled j in a random tree of size n.