Further combinatorial results for the symmetric inverse monoid
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)...
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| Дата: | 2022 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2022
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| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1793 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| Резюме: | 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 \(\mathcal{I}_{n}\). |
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