Some combinatorial problems in the theory of partial transformation semigroups

Let $X_n = \{1, 2, \ldots , n\}$. On a partial transformation $\alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow \mathop{\rm Im}\alpha \subseteq X_n$ of $X_n$ the following parameters are defined: the breadth or width of $\alpha$ is $\mid\mathop{\rm Dom}\nolimits \alpha\mid$, the collapse of $\alpha$ is $c(\alpha)=\mid\cup_{t \in \mathop{\rm Im}\alpha}\{t \alpha^{-1}: \mid t\alpha^{-1}\mid \geq 2\}\mid$, fix of $\alpha$ is $f(\alpha) = \mid\{x \in X_n: x\alpha = x\}\mid$, the height of $\alpha$ is $\mid\mathop{\rm Im}\alpha\mid$, and the right [left] waist of $\alpha$ is $\max(\mathop{\rm Im}\alpha)\, [\min(\mathop{\rm Im}\alpha)]$. The cardinalities of some equivalences defined by equalities of these parameters on $\mathcal{T}_n$, the semigroup of full transformations of $X_n$, and $\mathcal{P}_n$ the semigroup of partial transformations of $X_n$ and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted.