Summary: Let $ X_n = \{1, 2, \ldots , n\}$. On a partial transformation $ \alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow$ Im $ \alpha \subseteq X_n$ of $ X_n$ the following parameters are defined: the breadth or width of $ \alpha$ is $ \mid {\rm Dom}\ \alpha\mid$, the height of $ \alpha$ is $ \mid$ Im $ \alpha\mid$, and the right (resp., left) waist of $ \alpha$ is $ \max($Im $ \alpha)$ (resp., $ \min($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$.

20M18, 20M20, 05A10, 05A15

full transformation, partial transformation, partial one-to-one transformation, orientation-preserving transformation, orientation-reversing transformation, breadth, height, right waist, left waist, cyclic sequence, anti-cyclic sequence