# Forbidden Substrings In Circular K-Successions

## TL;DR

This paper studies permutations with circular k-successions, counting those avoiding specific substrings, and reveals that the counts depend on the primality and relative primality of n and k.

## Contribution

It introduces the concept of circular k-successions and provides formulas for counting permutations avoiding certain substrings based on number-theoretic properties.

## Key findings

- Number of permutations avoiding substrings depends on whether n is prime.
- Counts vary with the relative primality of n and k.
- Results connect permutation avoidance to number theory.

## Abstract

In this note we define circular k-successions in permutations in one-line notation and count permutations that avoid substrings j(j+k) and j(j+k) (mod n). We also count circular permutations that avoid such substrings, and show that for substrings j(j+k) (mod n), the number of permutations depends on whether n is prime, and more generally, on whether n and k are relatively prime.