  Let $n \ge 3$ be an integer. Let $P_n = \{1, 2, 3, ..., n-1, n \}$ and let
$S_n$ be the symmetric group of permutations on $P_n$. Motivated by the theory
of discrete dynamical systems on the interval, we associate each permutation
$\si_n$ in $S_n$ a (zero-one) Petrie matrix $M_{\si_n,n-1}$ in
$GL(n-1,{{\mathbb{R}}})$ (which is generally not the same as the usual
permutation matrix). Then, for any two permutations $\si_n$ and $\rho_n$ in
$S_n$, the notions of right, left and two-sided similarities (and weak
similarities respectively) of $\si_n$ and $\rho_n$ are introduced using the
similarities (and the characteristic polynomials respectively) of the
correspnding Petrie matrices of some extended permutations related to $\si_n$
and $\rho_n$ and examples are presented. As a by-product, we obtain ways to
construct countably infinitely many pairs of Petrie matrices which are similar.
