Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inversions in split trees and conditional Galton--Watson trees   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.

29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg