Further combinatorial results for the symmetric inverse monoid

Abstract

Let \(\mathcal{I}_{n}\) be the set of partial one-to-one transformations on the chain \(X_{n}=\{1,2,\dots,n\}\) and, for each \(\alpha\) in \(\mathcal{I}_{n}\), let \(h(\alpha)=|\operatorname{Im}\alpha|\), \(f(\alpha)=|\{x\in X_{n}\colon x\alpha=x\}|\) and \(w(\alpha) =\max(\operatorname{Im}\alpha) \). In this note, we obtain formulae involving binomial coefficients of \(F(n;p,m,k)=|\{\alpha\in\mathcal{I}_{n}\colon h(\alpha)=p\wedge f(\alpha)=m\wedge w(\alpha)=k\}|\) and \(F(n;\cdot,m,k)=|\{\alpha\in\mathcal{I}_{n}\colon f(\alpha)=m\wedge w(\alpha)=k\}|\) and analogous results on the set of partial derangements of \(\mathcal{I}_{n}\).