### 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 |
---|---|

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

**Combinatorial results for semigroups of order-preserving or order-reversing subpermutations.** / Laradji, A.; Umar, A.

Research output: Contribution to journal › Article

*Journal of Difference Equations and Applications*, vol. 21, no. 3, pp. 269-283. https://doi.org/10.1080/10236198.2015.1005080

}

TY - JOUR

T1 - Combinatorial results for semigroups of order-preserving or order-reversing subpermutations

AU - Laradji, A.

AU - Umar, A.

PY - 2015/3/4

Y1 - 2015/3/4

N2 - 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.

AB - 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.

KW - asymptotic behaviour

KW - generating function

KW - nilpotent transformation

KW - order-preserving and order-reversing subpermutations

KW - recurrence relation

UR - http://www.scopus.com/inward/record.url?scp=84924337176&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84924337176&partnerID=8YFLogxK

U2 - 10.1080/10236198.2015.1005080

DO - 10.1080/10236198.2015.1005080

M3 - Article

VL - 21

SP - 269

EP - 283

JO - Journal of Difference Equations and Applications

JF - Journal of Difference Equations and Applications

SN - 1023-6198

IS - 3

ER -