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\begin{center}
\vskip 1cm{\LARGE\bf Combinatorial Results for Semigroups of
\\ \vskip .05in
Orientation-Preserving Partial \\
\vskip .12in 
Transformations } \vskip 1cm \large
A. Umar \\
Department of Mathematics and Statistics\\
Sultan Qaboos University\\
Al-Khod, PC 123 \\
Sultanate of Oman \\
\href{mailto:aumarh@squ.edu.om}{\tt aumarh@squ.edu.om} \\
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\def\dom{\mathop{\rm Dom}\nolimits}

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\begin{abstract}
Let $X_n = \{1, 2, \ldots , n\}$.  On a partial transformation
$\alpha : \dom \alpha \subseteq  X_n \rightarrow \mbox{Im }\alpha
\subseteq X_n$ of $X_n$ the following parameters are defined: the
{\em breadth or width} of $\alpha$ is $\mid {\rm Dom}\ \alpha\mid$,
the {\em height} of $\alpha$ is $\mid \mbox{Im }\alpha\mid$, and the
{\em right (resp., left) waist} of $\alpha$ is $\max(\mbox{Im } \alpha)$
 (resp., $\min(\mbox{Im } \alpha)$). We compute the cardinalities of some
equivalences defined by equalities of these parameters on ${\cal
OP}_n$, the semigroup of orientation-preserving full transformations
of $X_n$, ${\cal POP}_n$ the semigroup of orientation-preserving
partial transformations of $X_n$, ${\cal OR}_n$ the semigroup of
orientation-preserving/reversing full transformations of $X_n$, and
${\cal POR}_n$ the semigroup of orientation-preserving/reversing
partial transformations of $X_n$, and their partial one-to-one
analogue semigroups, ${\cal POPI}_n$ and ${\cal PORI}_n$.
\end{abstract}


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