# Permutation Matrices, Their Discrete Derivatives and Extremal Properties

## TL;DR

This paper introduces the concept of discrete derivatives for permutation matrices, characterizes their possible values, and explores special classes like permutations with distinct derivatives and their connection to Costas arrays.

## Contribution

It defines and characterizes the discrete derivatives of permutations and investigates their properties and applications, including relations to Costas arrays.

## Key findings

- Characterization of possible derivatives of permutations
- Identification of permutations with distinct derivatives
- Connection between derivatives and Costas arrays

## Abstract

For a permutation $\pi$, and the corresponding permutation matrix, we introduce the notion of {\em discrete derivative}, obtained by taking differences of successive entries in $\pi$. We characterize the possible derivatives of permutations, and consider questions for permutations with certain properties satisfied by the derivative. For instance, we consider permutations with distinct derivatives, and the relationship to so-called Costas arrays.