# Permuting Operations on Strings: Their Permutations and Their Primes

P.R.J. Asveld

Research output: Book/ReportReportProfessional

### Abstract

We study some length-preserving operations on strings that permute the symbol positions in strings. These operations include some well-known examples (reversal, circular or cyclic shift, shuffle, twist, operations induced by the Josephus problem) and some new ones based on the Archimedes spiral. Such a permuting 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 such a permutation $X_n$. We call an integer $n$ $X$-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.
Original language Undefined Enschede Human Media Interaction (HMI) 46 Published - 30 Jun 2009

### Publication series

Name CTIT Technical Report Series University of Twente, Centre for Telematica and Information Technology (CTIT) TR-CTIT-09-26 1381-3625

### Keywords

• Josephus problem
• Shuffle
• permutation
• Operation on strings
• cyclic subgroup
• EWI-15655
• distribution of prime numbers
• prime number
• MSC-68R15
• HMI-SLT: Speech and Language Technology
• MSC-11A07
• Twist
• IR-67513
• MSC-11A41
• METIS-263904

### Cite this

Asveld, P. R. J. (2009). Permuting Operations on Strings: Their Permutations and Their Primes. (CTIT Technical Report Series; No. TR-CTIT-09-26). Enschede: Human Media Interaction (HMI).
Asveld, P.R.J. / Permuting Operations on Strings: Their Permutations and Their Primes. Enschede : Human Media Interaction (HMI), 2009. 46 p. (CTIT Technical Report Series; TR-CTIT-09-26).
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title = "Permuting Operations on Strings: Their Permutations and Their Primes",
abstract = "We study some length-preserving operations on strings that permute the symbol positions in strings. These operations include some well-known examples (reversal, circular or cyclic shift, shuffle, twist, operations induced by the Josephus problem) and some new ones based on the Archimedes spiral. Such a permuting 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 such a permutation $X_n$. We call an integer $n$ $X$-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.",
keywords = "Josephus problem, Shuffle, permutation, Operation on strings, cyclic subgroup, EWI-15655, distribution of prime numbers, prime number, MSC-68R15, HMI-SLT: Speech and Language Technology, MSC-11A07, Twist, IR-67513, MSC-11A41, METIS-263904",
author = "P.R.J. Asveld",
note = "eemcs-eprint-15655",
year = "2009",
month = "6",
day = "30",
language = "Undefined",
series = "CTIT Technical Report Series",
publisher = "Human Media Interaction (HMI)",
number = "TR-CTIT-09-26",

}

Asveld, PRJ 2009, Permuting Operations on Strings: Their Permutations and Their Primes. CTIT Technical Report Series, no. TR-CTIT-09-26, Human Media Interaction (HMI), Enschede.
Enschede : Human Media Interaction (HMI), 2009. 46 p. (CTIT Technical Report Series; No. TR-CTIT-09-26).

Research output: Book/ReportReportProfessional

TY - BOOK

T1 - Permuting Operations on Strings: Their Permutations and Their Primes

AU - Asveld, P.R.J.

N1 - eemcs-eprint-15655

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Y1 - 2009/6/30

N2 - We study some length-preserving operations on strings that permute the symbol positions in strings. These operations include some well-known examples (reversal, circular or cyclic shift, shuffle, twist, operations induced by the Josephus problem) and some new ones based on the Archimedes spiral. Such a permuting 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 such a permutation $X_n$. We call an integer $n$ $X$-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.

AB - We study some length-preserving operations on strings that permute the symbol positions in strings. These operations include some well-known examples (reversal, circular or cyclic shift, shuffle, twist, operations induced by the Josephus problem) and some new ones based on the Archimedes spiral. Such a permuting 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 such a permutation $X_n$. We call an integer $n$ $X$-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.

KW - Josephus problem

KW - Shuffle

KW - permutation

KW - Operation on strings

KW - cyclic subgroup

KW - EWI-15655

KW - distribution of prime numbers

KW - prime number

KW - MSC-68R15

KW - HMI-SLT: Speech and Language Technology

KW - MSC-11A07

KW - Twist

KW - IR-67513

KW - MSC-11A41

KW - METIS-263904

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BT - Permuting Operations on Strings: Their Permutations and Their Primes

PB - Human Media Interaction (HMI)

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ER -

Asveld PRJ. Permuting Operations on Strings: Their Permutations and Their Primes. Enschede: Human Media Interaction (HMI), 2009. 46 p. (CTIT Technical Report Series; TR-CTIT-09-26).