  Inspired by the Stochastic processes described by the Feller Coupling and
Chinese Restaurant Processes, we create four different bijections from words in
the set $[1]\times [2] \times\cdot \times[n]$ to $S_n$. We then compose these
maps with their inverse to obtain a toal of six bijections $S_n \to S_n$.
Following that, we investigate the fixed points ($1$-cycle) and higher
$k$-cycles of these maps. We characterized some of their properties completely
as well as empirically showing the complexity of the higher $k$-cycle
structures for these maps.
