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: | Англійська |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2022
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| 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| _version_ | 1856543055000109056 |
|---|---|
| author | Laradji, A. Umar, A. |
| author_facet | Laradji, A. Umar, A. |
| author_sort | Laradji, A. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2022-10-14T16:01:17Z |
| description | 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 \(\mathcal{I}_{n}\). |
| first_indexed | 2025-12-02T15:30:33Z |
| format | Article |
| id | admjournalluguniveduua-article-1793 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:30:33Z |
| publishDate | 2022 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-17932022-10-14T16:01:17Z Further combinatorial results for the symmetric inverse monoid Laradji, A. Umar, A. 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 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 \(\mathcal{I}_{n}\). Lugansk National Taras Shevchenko University 2022-10-14 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1793 10.12958/adm1793 Algebra and Discrete Mathematics; Vol 33, No 2 (2022) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1793/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1793/854 Copyright (c) 2022 Algebra and Discrete Mathematics |
| spellingShingle | 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 Laradji, A. Umar, A. Further combinatorial results for the symmetric inverse monoid |
| title | Further combinatorial results for the symmetric inverse monoid |
| title_full | Further combinatorial results for the symmetric inverse monoid |
| title_fullStr | Further combinatorial results for the symmetric inverse monoid |
| title_full_unstemmed | Further combinatorial results for the symmetric inverse monoid |
| title_short | Further combinatorial results for the symmetric inverse monoid |
| title_sort | further combinatorial results for the symmetric inverse monoid |
| topic | 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 |
| topic_facet | 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 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1793 |
| work_keys_str_mv | AT laradjia furthercombinatorialresultsforthesymmetricinversemonoid AT umara furthercombinatorialresultsforthesymmetricinversemonoid |