Combinatorial differential operators in: Fa\`a di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees

TL;DR

This paper develops differential operator techniques to derive formulas and bijections for enumerating increasing trees, ballot paths, and Dyck paths, connecting combinatorics with differential equations and formal power series.

Contribution

It introduces a differential equation approach for counting increasing trees and provides a bijective proof of a Faà di Bruno formula variant, unifying several combinatorial structures.

Findings

Derived a differential equation for path length enumeration in increasing trees

Provided a bijective proof of a Faà di Bruno formula variant

Formulated recursive and explicit formulas for enriched trees and paths

Abstract

We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Fa\`a di Bruno formula, and model the generation of ballot and Dyck paths. We get formulas for its enumeration according with the height of their lattice points. Recursive formulas for the enumeration of enriched increasing trees and forests with respect to the height of their internal and external vertices are also obtained. Finally we present a generalized form of all those results using one-parameter groups in the general context of formal power series in an arbitrary number of variables.