Inversions in split trees and conditional Galton--Watson trees

TL;DR

This paper investigates the distribution of inversions in various random tree models, deriving explicit formulas and convergence results, including limits involving Brownian excursions, thereby extending previous work on tree inversions.

Contribution

It provides explicit cumulant formulas for inversions, establishes convergence conditions for normalized inversions, and identifies limit distributions for split and Galton--Watson trees, broadening the understanding of tree inversion behavior.

Findings

Explicit cumulant formulas involving common ancestors.

Convergence conditions for normalized inversions in fixed trees.

Limit distribution for Galton--Watson trees related to Brownian excursions.

Abstract

We study $I(T)$, the number of inversions in a tree $T$ with its vertices labeled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of $I(T)$ have explicit formulas involving the $k$-total common ancestors of $T$ (an extension of the total path length). Then we consider $X_n$, the normalized version of $I(T_n)$, for a sequence of trees $T_n$. For fixed $T_{n}$'s, we prove a sufficient condition for $X_n$ to converge in distribution. As an application, we identify the limit of $X_n$ for complete $b$-ary trees. For $T_n$ being split trees, we show that $X_n$ converges to the unique solution of a distributional equation. Finally, when $T_n$'s are conditional Galton--Watson trees, we show that $X_n$ converges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that are stronger and much broader compared to previous work by Panholzer and Seitz.