### Abstract

Let X_{n} = {1, 2,..., n}. On a partial transformation α: Dom α ⊆ Xn! Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is {pipe} Dom α {pipe}, the height of α is {pipe} Im α {pipe}, and the right (resp., left) waist of α is max(Im α) (resp., min(Im α)). We compute the cardinalities of some equivalences defined by equalities of these parameters on OP_{n}, the semigroup of orientation-preserving full transformations of X_{n}, POP_{n} the semigroup of orientation-preserving partial transformations of X_{n}, OR_{n} the semigroup of orientation-preserving/reversing full transformations of X_{n}, and POR_{n} the semigroup of orientation-preserving/reversing partial transformations of X_{n}, and their partial one-to-one analogue semigroups, POPI_{n} and PORI_{n}.

Original language | English |
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Journal | Journal of Integer Sequences |

Volume | 14 |

Issue number | 7 |

Publication status | Published - 2011 |

### Keywords

- Anti-cyclic sequence
- Breadth
- Cyclic sequence
- Full transformation
- Height
- Left waist
- Orientation-preserving transformation
- Orientation-reversing transformation
- Partial one-to-one transformation
- Partial transformation
- Right waist

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics