Further combinatorial results for the symmetric inverse monoid

• Date: 2022
• Main Authors: Laradji, A., Umar, A.
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2022
• Subjects: partial one-to-one transformation; symmetric inverse monoid; height of \(\alpha\); fix of \(\alpha\); (left) waist of \(\alpha\); permutation; (partial) derangement; 20M18; 20M20; 05A10; 05A15
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1793
• Journal Title: Algebra and Discrete Mathematics

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)...