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

Abstract

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 $ρ_n$ in $S_n$, the notions of right, left and two-sided similarities (and weak similarities respectively) of $\si_n$ and $ρ_n$ are introduced using the similarities (and the characteristic polynomials respectively) of the correspnding Petrie matrices of some extended permutations related to $\si_n$ and $ρ_n$ and examples are presented. As a by-product, we obtain ways to construct countably infinitely many pairs of Petrie matrices which are similar.