Let
![$ X_n = \{1, 2, \ldots , n\}$](abs/img2.gif)
. On a partial transformation
![$ \alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow$](abs/img3.gif)
Im
![$ \alpha
\subseteq X_n$](abs/img4.gif)
of
![$ X_n$](abs/img5.gif)
the following parameters are defined: the
breadth or width of
![$ \alpha$](abs/img6.gif)
is
![$ \mid {\rm Dom}\ \alpha\mid$](abs/img7.gif)
,
the
height of
![$ \alpha$](abs/img6.gif)
is
![$ \mid$](abs/img8.gif)
Im
![$ \alpha\mid$](abs/img9.gif)
, and the
right (resp., left) waist of
![$ \alpha$](abs/img6.gif)
is
![$ \max($](abs/img10.gif)
Im
![$ \alpha)$](abs/img11.gif)
(resp.,
![$ \min($](abs/img12.gif)
Im
![$ \alpha)$](abs/img11.gif)
). We compute the cardinalities of some
equivalences defined by equalities of these parameters on
![$ {\cal
OP}_n$](abs/img13.gif)
, the semigroup of orientation-preserving full transformations
of
![$ X_n$](abs/img5.gif)
,
![$ {\cal POP}_n$](abs/img14.gif)
the semigroup of orientation-preserving
partial transformations of
![$ X_n$](abs/img5.gif)
,
![$ {\cal OR}_n$](abs/img15.gif)
the semigroup of
orientation-preserving/reversing full transformations of
![$ X_n$](abs/img5.gif)
, and
![$ {\cal POR}_n$](abs/img16.gif)
the semigroup of orientation-preserving/reversing
partial transformations of
![$ X_n$](abs/img5.gif)
, and their partial one-to-one
analogue semigroups,
![$ {\cal POPI}_n$](abs/img17.gif)
and
![$ {\cal PORI}_n$](abs/img18.gif)
.