# Permuting Operations on Strings and Their Relation to Prime Numbers

P.R.J. Asveld

Research output: Book/ReportReportProfessional

### Abstract

Some length-preserving operations on strings only permute the symbol positions in strings; such an operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We investigate the structure and the order of the cyclic group generated by $X_n$. We call an integer $n$ $X$-{\em prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). 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 Undefined Enschede Centre for Telematics and Information Technology (CTIT) 24 Published - 7 Jun 2010

### Publication series

Name CTIT Technical Report Series University of Twente, Centre for Telematica and Information Technology (CTIT) TR-CTIT-10-22 1381-3625

### Keywords

• Operation on strings
• MSC-11A41
• Josephus problem
• Queneau number
• Twist
• IR-71704
• METIS-270843
• Shuffle
• MSC-11A07
• HMI-SLT: Speech and Language Technology
• MSC-68R15
• prime number
• EWI-17983

### Cite this

Asveld, P. R. J. (2010). Permuting Operations on Strings and Their Relation to Prime Numbers. (CTIT Technical Report Series; No. TR-CTIT-10-22). Enschede: Centre for Telematics and Information Technology (CTIT).
Asveld, P.R.J. / Permuting Operations on Strings and Their Relation to Prime Numbers. Enschede : Centre for Telematics and Information Technology (CTIT), 2010. 24 p. (CTIT Technical Report Series; TR-CTIT-10-22).
@book{5f398dc657934d00b6a8f799d7475211,
title = "Permuting Operations on Strings and Their Relation to Prime Numbers",
abstract = "Some length-preserving operations on strings only permute the symbol positions in strings; such an operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We investigate the structure and the order of the cyclic group generated by $X_n$. We call an integer $n$ $X$-{\em prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). 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.",
keywords = "Operation on strings, MSC-11A41, Josephus problem, Queneau number, Twist, IR-71704, METIS-270843, Shuffle, MSC-11A07, HMI-SLT: Speech and Language Technology, MSC-68R15, prime number, EWI-17983",
author = "P.R.J. Asveld",
note = "eemcs-eprint-17983",
year = "2010",
month = "6",
day = "7",
language = "Undefined",
series = "CTIT Technical Report Series",
publisher = "Centre for Telematics and Information Technology (CTIT)",
number = "TR-CTIT-10-22",

}

Asveld, PRJ 2010, Permuting Operations on Strings and Their Relation to Prime Numbers. CTIT Technical Report Series, no. TR-CTIT-10-22, Centre for Telematics and Information Technology (CTIT), Enschede.
Enschede : Centre for Telematics and Information Technology (CTIT), 2010. 24 p. (CTIT Technical Report Series; No. TR-CTIT-10-22).

Research output: Book/ReportReportProfessional

TY - BOOK

T1 - Permuting Operations on Strings and Their Relation to Prime Numbers

AU - Asveld, P.R.J.

N1 - eemcs-eprint-17983

PY - 2010/6/7

Y1 - 2010/6/7

N2 - Some length-preserving operations on strings only permute the symbol positions in strings; such an operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We investigate the structure and the order of the cyclic group generated by $X_n$. We call an integer $n$ $X$-{\em prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). 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.

AB - Some length-preserving operations on strings only permute the symbol positions in strings; such an operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We investigate the structure and the order of the cyclic group generated by $X_n$. We call an integer $n$ $X$-{\em prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). 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.

KW - Operation on strings

KW - MSC-11A41

KW - Josephus problem

KW - Queneau number

KW - Twist

KW - IR-71704

KW - METIS-270843

KW - Shuffle

KW - MSC-11A07

KW - HMI-SLT: Speech and Language Technology

KW - MSC-68R15

KW - prime number

KW - EWI-17983

M3 - Report

T3 - CTIT Technical Report Series

BT - Permuting Operations on Strings and Their Relation to Prime Numbers

PB - Centre for Telematics and Information Technology (CTIT)

CY - Enschede

ER -

Asveld PRJ. Permuting Operations on Strings and Their Relation to Prime Numbers. Enschede: Centre for Telematics and Information Technology (CTIT), 2010. 24 p. (CTIT Technical Report Series; TR-CTIT-10-22).