A study on the fixed points of the $\gamma$ function

TL;DR

This paper investigates the fixed points of the $oldsymbol{ extgamma}$-operator on Dyck paths, providing new combinatorial insights and an algorithm to generate all fixed points of a given degree.

Contribution

It introduces new properties of fixed points of $oldsymbol{ extgamma}$, and presents an algorithm to generate all fixed points of a specified degree.

Findings

New combinatorial properties of $oldsymbol{ extgamma}$ fixed points

An algorithm to generate fixed points of a given degree

Enhanced understanding of the structure of fixed points

Abstract

Recently a permutation on Dyck paths, related to the chip firing game, was introduced and studied by Barnabei et al.. It is called $\gamma$-operator, and uses symmetries and reflections to relate Dyck paths having the same length. A relevant research topic concerns the study of the fixed points of $\gamma$ and a characterization of these objects was provided by Barnabei et al, leaving the problem of their enumeration open. In this paper, using tools from combinatorics of words, we determine new combinatorial properties of the fixed points of $\gamma$. Then we present an algorithm, denoted by \textbf{GenGammaPath}($t$), which receives as input an array $t=(t_0, \ldots ,t_{k})$ of positive integers and generates all the elements of $F_{\gamma}$ with degree $k$.