Runs in labelled trees and mappings

TL;DR

This paper extends the concept of ascending and descending runs from permutations to rooted labelled trees and mappings, providing generating functions, enumeration formulas, and bijections to analyze these runs.

Contribution

It introduces a unified combinatorial framework for runs in trees and mappings, including generating functions, limit theorems, and explicit formulas with Stirling numbers.

Findings

Derived bivariate central limit theorems for runs

Obtained explicit enumeration formulas involving Stirling numbers

Established bijections between mappings and labelled trees

Abstract

We generalize the concept of ascending and descending runs from permutations to rooted labelled trees and mappings, i.e., functions from the set $\{1, \dots, n\}$ into itself. A combinatorial decomposition of the corresponding functional digraph together with a generating functions approach allows us to perform a joint study of ascending and descending runs in labelled trees and mappings, respectively. From the given characterization of the respective generating functions we can deduce bivariate central limit theorems for these quantities. Furthermore, for ascending runs (or descending runs) we gain explicit enumeration formulae showing a connection to Stirling numbers of the second kind. We also give a bijective proof establishing this relation, and further state a bijection between mappings and labelled trees connecting the quantities in both structures.