arXiv math/01120182001-12-03math.CO

包含和避免某些模式的着色排列

Coloured permutations containing and avoiding certain patterns

T. Mansour

备注：: 7 pages

摘要：

按照曼苏尔的说法，让$S_n^{(r)}$是符号$1,2,...,n$上所有颜色排列的集合，颜色为$1,2,...,r$，当r=1时，它类似于对称群，当r=2时，它类似于超八面体群。设$I\subseteq\{1,2,...,r\}$是d种颜色的子集，我们定义$T_{k,r}^m(I)$是所有着色排列的集合，$ϕ\in S_k^{(r)}$使得$ϕ_1=m^{(c)}$，其中$c\in I$。证明了$S_n^{(r)}$中$T_{k,r}^m(I)$ -避免色置换的个数等于$(k-1)!r^{k-1}\prod_{j=k}^n h_j$，其中$h_j=(r-d)j+(k-1)d$为$n\geq k$.然后，我们证明，对于任何$ϕ\in T_{k,r}^1(I)$（或任何$ϕ\in T_{k,r}^k(I)$），在$S_n^{(r)}$中的彩色排列的数量，避免所有的模式在$T_{k,r}^1(I)$（或在$T_{k,r}^k(I)$）除了$ϕ$和包含$ϕ$正好一次等于$\prod_{j=k}^n h_j\cdot \sum_{j=k}^n \frac{1}{h_j}$的$n\geq k$。最后，对于任何$ϕ\in T_{k,r}^m(I)$，$2\leq m\leq k-1$，该数字等于$n\geq k+1$的$\prod_{j=k+1}^n h_j$。这些结果推广了Mansour和Simion最近的结果。

英文摘要：

Following Mansour, let $S_n^{(r)}$ be the set of all coloured permutations on the symbols $1,2,...,n$ with colours $1,2,...,r$, which is the analogous of the symmetric group when r=1, and the hyperoctahedral group when r=2. Let $I\subseteq\{1,2,...,r\}$ be subset of d colours; we define $T_{k,r}^m(I)$ be the set of all coloured permutations $ϕ\in S_k^{(r)}$ such that $ϕ_1=m^{(c)}$ where $c\in I$. We prove that, the number $T_{k,r}^m(I)$-avoiding coloured permutations in $S_n^{(r)}$ equals $(k-1)!r^{k-1}\prod_{j=k}^n h_j$ for $n\geq k$ where $h_j=(r-d)j+(k-1)d$. We then prove that for any $ϕ\in T_{k,r}^1(I)$ (or any $ϕ\in T_{k,r}^k(I)$), the number of coloured permutations in $S_n^{(r)}$ which avoid all patterns in $T_{k,r}^1(I)$ (or in $T_{k,r}^k(I)$) except for $ϕ$ and contain $ϕ$ exactly once equals $\prod_{j=k}^n h_j\cdot \sum_{j=k}^n \frac{1}{h_j}$ for $n\geq k$. Finally, for any $ϕ\in T_{k,r}^m(I)$, $2\leq m\leq k-1$, this number equals $\prod_{j=k+1}^n h_j$ for $n\geq k+1$. These results generalize recent results due to Mansour, and due to Simion.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文