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

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