# Permuting Operations on Strings and Their Relation to Prime Numbers

P.R.J. Asveld

3 Citations (Scopus)

## Abstract

Some length-preserving operations on strings only permute the symbol positions in strings; such an operation $X$ gives rise to a family $\{X_n\}_{n\geq 2}$ of similar permutations. We investigate the structure and the order of the cyclic group generated by $X_n$. We call an integer $n$ $X$-prime if $X_n$ consists of a single cycle of length $n$ ($n\geq 2$). Then we show some properties of these $X$-primes, particularly, how $X$-primes are related to $X^\prime$-primes as well as to ordinary prime numbers. Here $X$ and $X^\prime$ range over well-known examples (reversal, cyclic shift, shuffle, twist) and some new ones based on the Archimedes spiral and on the Josephus problem.
Original language English 1915-1932 18 Discrete applied mathematics 159 17 https://doi.org/10.1016/j.dam.2011.07.019 Published - 28 Oct 2011

## Keywords

• Operation on strings
• prime number
• MSC-68R15
• HMI-SLT: Speech and Language Technology
• MSC-11A07
• MSC-12E20
• Twist
• Queneau number
• Shuffle
• Josephus problem