### 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 |
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Place of Publication | Enschede |

Publisher | Human Media Interaction (HMI) |

Number of pages | 46 |

Publication status | Published - 30 Jun 2009 |

### Publication series

Name | CTIT Technical Report Series |
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Publisher | University of Twente, Centre for Telematica and Information Technology (CTIT) |

No. | TR-CTIT-09-26 |

ISSN (Print) | 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).