Connections between certain numbers related to derangements and $r$-permutations
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| Format: | Preprint |
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2024
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| _version_ | 1866911269213175808 |
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| author | Miska, Piotr Żmija, Błażej |
| author_facet | Miska, Piotr Żmija, Błażej |
| contents | For non-negative integer parameters $r,u,m,n$ define \begin{align*}
\cal{D}(r,u,m,n) := \big\{\ σ\in \cal{S}_{r+n}\ \big|\ σ(x)=y \textrm{ for exactly } u \textrm{ pairs } (x,y) \textrm{ such that } 1\leq x,y\leq r \textrm{ and } σ(t)=t \textrm{ for exactly } m \textrm{ elements } r+1\leq t\leq r+n\ \big\} \end{align*} and \begin{align*}
\cal{D}_{r,u,m}(n) := \big\{\ σ\in \cal{S}_{r+n}\ \big|\ \forall_{1\leq x<y\leq r} \ x \textrm{ and } y \textrm{ are in disjoint cycles of } σ\textrm{ and } σ(z)=z \textrm{ for exactly } u \textrm{ elements } 1\leq z\leq r, \textrm{ and } σ(t)=t \textrm{ for exactly } m \textrm{ elements } r+1\leq t\leq r+n\ \big\}, \end{align*} where $\mathcal{S}_{n}$ denotes the set of all the permutations of $\{1,\ldots ,n\}$.
In this paper we study connections between the sets $\mathcal{D}(r,u,m,n)$, $\mathcal{D}_{r,u,m}(n)$, and the sets of (some classes of) $r$-derangements. We rely mostly on counting arguments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19294 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connections between certain numbers related to derangements and $r$-permutations Miska, Piotr Żmija, Błażej Combinatorics Number Theory 11B75 (Primary) 11B37 (Secondary) For non-negative integer parameters $r,u,m,n$ define \begin{align*} \cal{D}(r,u,m,n) := \big\{\ σ\in \cal{S}_{r+n}\ \big|\ σ(x)=y \textrm{ for exactly } u \textrm{ pairs } (x,y) \textrm{ such that } 1\leq x,y\leq r \textrm{ and } σ(t)=t \textrm{ for exactly } m \textrm{ elements } r+1\leq t\leq r+n\ \big\} \end{align*} and \begin{align*} \cal{D}_{r,u,m}(n) := \big\{\ σ\in \cal{S}_{r+n}\ \big|\ \forall_{1\leq x<y\leq r} \ x \textrm{ and } y \textrm{ are in disjoint cycles of } σ\textrm{ and } σ(z)=z \textrm{ for exactly } u \textrm{ elements } 1\leq z\leq r, \textrm{ and } σ(t)=t \textrm{ for exactly } m \textrm{ elements } r+1\leq t\leq r+n\ \big\}, \end{align*} where $\mathcal{S}_{n}$ denotes the set of all the permutations of $\{1,\ldots ,n\}$. In this paper we study connections between the sets $\mathcal{D}(r,u,m,n)$, $\mathcal{D}_{r,u,m}(n)$, and the sets of (some classes of) $r$-derangements. We rely mostly on counting arguments. |
| title | Connections between certain numbers related to derangements and $r$-permutations |
| topic | Combinatorics Number Theory 11B75 (Primary) 11B37 (Secondary) |
| url | https://arxiv.org/abs/2411.19294 |