Some combinatorial problems in the theory of symmetric inverse semigroups

Let $X_n =\{1, 2,\cdots,n\}$ and let $\alpha:\operatorname{Dom}\alpha\subseteq X_n\rightarrow\operatorname{Im}\alpha\subseteq X_n$ be a (partial) transformation on $X_n$. On a partial one-one mapping of $X_n$ the following parameters are defined: the height of $\alpha$ is $h(\alpha)=|\operatorname{Im}\alpha|$, the right [left] waist of $\alpha$ is $w^+(\alpha)=\max(\operatorname{Im}\alpha)[w^-(\alpha)=\min(\operatorname{Im}\alpha)]$, and fix of $\alpha$ is denoted by $f(\alpha)$, and defined by $f(\alpha)=|\{x\in X_n:x\alpha=x\}|$. The cardinalities of some equivalences defined by equalities of these parameters on ${\mathcal I}_n$, the semigroup of partial one-one mappings of $X_n$, and some of its notable subsemigroups that have been computed are gathered together and the open problems highlighted.