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

TL;DR

This paper introduces new concepts of one-sided and two-sided similarities between permutations based on associated Petrie matrices, linking permutation theory with matrix similarity and dynamical systems.

Contribution

It defines novel similarity notions for permutations using Petrie matrices and explores their properties, including methods to generate infinitely many similar matrix pairs.

Findings

Defined right, left, and two-sided similarities for permutations.

Connected permutation similarities to matrix characteristic polynomials.

Provided examples and methods to construct infinitely many similar Petrie matrix pairs.

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 $\rho_n$ 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.