Logarithms of Catalan generating functions: A combinatorial approach

TL;DR

This paper provides a combinatorial interpretation of the logarithm of generalized Catalan generating functions, using lattice paths and tree graphs, and extends the approach to higher powers.

Contribution

It introduces a new combinatorial approach to analyze the logarithm of Catalan generating functions and generalizes to higher powers using explicit bijections.

Findings

Combinatorial interpretations for log G_k via lattice paths and trees

Explicit bijections recover known coefficients

Generalization to higher powers of logarithm

Abstract

We analyze the combinatorics behind the operation of taking the logarithm of the generating function $G_k$ for $k^\text{th}$ generalized Catalan numbers. We provide combinatorial interpretations in terms of lattice paths and in terms of tree graphs. Using explicit bijections, we are able to recover known closed expressions for the coefficients of $\log G_k$ by purely combinatorial means of enumeration. The non-algebraic proof easily generalizes to higher powers $\log^a G_k$, $a\geq 2$.