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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 theoryof 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 usualpermutation matrix). Then, for any two permutations $\si_n$ and $\rho_n$ in$S_n$, the notions of right, left and two-sided similarities (and weaksimilarities respectively) of $\si_n$ and $\rho_n$ are introduced using thesimilarities (and the characteristic polynomials respectively) of thecorrespnding Petrie matrices of some extended permutations related to $\si_n$and $\rho_n$ and examples are presented. As a by-product, we obtain ways toconstruct countably infinitely many pairs of Petrie matrices which are similar.

On the one-sided and two-sided similarities or weak similarities of permutations