  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 $k$.
