### Abstract

Let (Formula presented.) For a partial one–one transformation (or subpermutation) (Formula presented.) of (Formula presented.) the following parameters are defined: the height (Formula presented.) , the waist (Formula presented.) , and the fix (Formula presented.). We compute the cardinalities of some equivalence classes defined by equalities of these parameters on (Formula presented.) and (Formula presented.) the semigroups of order-preserving and of order-preserving or order-reversing subpermutations of (Formula presented.) respectively. As a consequence, we obtain several formulae and generating functions for the number of nilpotents in (Formula presented.) and (Formula presented.). We also prove that, for large (Formula presented.) a randomly chosen order-preserving (resp. order-reversing) subpermutation of (Formula presented.) has probability (Formula presented.) (resp. (Formula presented.) ) of being nilpotent.

Original language | English |
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Pages (from-to) | 269-283 |

Number of pages | 15 |

Journal | Journal of Difference Equations and Applications |

Volume | 21 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 4 2015 |

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### Keywords

- asymptotic behaviour
- generating function
- nilpotent transformation
- order-preserving and order-reversing subpermutations
- recurrence relation

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics
- Analysis

### Cite this

*Journal of Difference Equations and Applications*,

*21*(3), 269-283. https://doi.org/10.1080/10236198.2015.1005080