Fixed points of orientation-preserving full transformation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866909000801452032 |
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| author | An, Yang Zhang, Wen Ting He, Yi |
| author_facet | An, Yang Zhang, Wen Ting He, Yi |
| contents | Let $\mathcal{OP}_n$ be the monoid of all orientation-preserving full transformations on $X_n=\{1,\dots, n\}$ with the natural order. For $α\in \mathcal{OP}_n$, let $F(α)=\{y\in X_n: yα=y\}$ and $F(n,m)=|\{α:|F(α)|=m\}|$. Umar posed the question about the number $F(n,m)$ of elements of $\mathcal{OP}_n$ with $m$ fixed points. In this paper, we show that the number $F(n,m)$ of $\mathcal{OP}_n$ is $\binom{2n}{n-m}$ for $2\leqslant m\leqslant n$ and get the expectation and probability distribution of the cardinality of fixed-point set $F(α)$ for $α\in\mathcal{OP}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26661 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fixed points of orientation-preserving full transformation An, Yang Zhang, Wen Ting He, Yi Group Theory Let $\mathcal{OP}_n$ be the monoid of all orientation-preserving full transformations on $X_n=\{1,\dots, n\}$ with the natural order. For $α\in \mathcal{OP}_n$, let $F(α)=\{y\in X_n: yα=y\}$ and $F(n,m)=|\{α:|F(α)|=m\}|$. Umar posed the question about the number $F(n,m)$ of elements of $\mathcal{OP}_n$ with $m$ fixed points. In this paper, we show that the number $F(n,m)$ of $\mathcal{OP}_n$ is $\binom{2n}{n-m}$ for $2\leqslant m\leqslant n$ and get the expectation and probability distribution of the cardinality of fixed-point set $F(α)$ for $α\in\mathcal{OP}_n$. |
| title | Fixed points of orientation-preserving full transformation |
| topic | Group Theory |
| url | https://arxiv.org/abs/2604.26661 |