Some combinatorial problems in the theory of symmetric inverse semigroups

Abstract

Let \(X_n = \{1, 2, \cdots , n\}\) and let \(\alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq  X_n \rightarrow \mathop{\rm Im}\nolimits \alpha \subseteq X_n\) 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  \(h(\alpha)=|\mathop{\rm Im}\nolimits \alpha|\), the right [left] waist of \(\alpha\) is \(w^+(\alpha) = \max(\mathop{\rm Im}\nolimits \alpha)[w^-(\alpha) = \min(\mathop{\rm Im}\nolimits \alpha)]\),  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.