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 $\log^a G_k$, $a\geq 2$.
@article{10_37236_11855,
author = {Sabine Jansen and Leonid Kolesnikov},
title = {Logarithms of {Catalan} generating functions: a combinatorial approach},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {1},
doi = {10.37236/11855},
zbl = {1535.05018},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11855/}
}
TY - JOUR
AU - Sabine Jansen
AU - Leonid Kolesnikov
TI - Logarithms of Catalan generating functions: a combinatorial approach
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11855/
DO - 10.37236/11855
ID - 10_37236_11855
ER -
%0 Journal Article
%A Sabine Jansen
%A Leonid Kolesnikov
%T Logarithms of Catalan generating functions: a combinatorial approach
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11855/
%R 10.37236/11855
%F 10_37236_11855
Sabine Jansen; Leonid Kolesnikov. Logarithms of Catalan generating functions: a combinatorial approach. The electronic journal of combinatorics, Tome 31 (2024) no. 1. doi: 10.37236/11855
