Generalized K-Shift Forbidden Substrings in Permutations

TL;DR

This paper studies permutations avoiding certain forbidden substrings, extending previous work to include modular conditions, with results depending on properties like primality and coprimality of n and k.

Contribution

It generalizes the analysis of forbidden substring permutations to modular cases, revealing dependence on number-theoretic properties such as primality and coprimality.

Findings

Number of permutations from the non-modular case derived via recursions from derangement numbers.

In the modular case, permutation counts depend on whether n is prime.

Permutation counts are influenced by the coprimality of n and k.

Abstract

In this note we continue the analysis of permutations that avoid substrings j(j+k), 1 <= j <= n-k, k < n, as well as substrings j(j+k) (mod n), 1 <= j <= n. In the first case the number of such permutations can be obtained from recursions starting from derangement numbers, while in the (mod n) case the number of permutations depends on whether n is prime, and more generally, on whether n and k are relatively prime.