Let
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$$ s^{(r)}_{n} $$
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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
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$$ I \subseteq \{1,2,\ldots,r\} $$
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be a subset of d colours; we define
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$$ T_{k,r}^m (l) $$
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to be the set of all coloured permutations
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$$ \Phi \in S_{k}^{(r)} \quad \mathrm{such\quad that}\quad \Phi_1 =
M^{(c)} \quad \mathrm{where}\quad c \in I$$
\end{document}.
We prove that the number of
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$$ T_{k,r}^{m} (l) $$
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-avoiding coloured permutations in
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$$ S_{n}^{(r)} \quad \mathrm{equals}\quad (k-1)!r^{k-1}
\prod_{j=k}^n h_j \quad \mathrm{for}\quad n\geq k \quad
\mathrm{where} \quad h_j = (r-d) j+(k-1)d $$
\end{document}.
We then prove that for any
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$$ \Phi \in T_{k,r}^{1} (I) \quad(\mathrm{or \quad any}\quad \Phi \in
T_{k,r}^{k} (I)) $$
\end{document},
the number of coloured permutations in
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$$ S-{n}^{(r)} $$
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which avoid all patterns in
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$$ T_{k,r}^{1} (I) \quad(\mathrm{or \quad in}\quad T_{k,r}^{k} (I)) $$
\end{document}
except for Φ and contain Φ exactly once equals
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$$ \prod_{j=k}^n h_j \cdot \sum_{j=k}^{n} \frac{1}{hj \quad}
\mathrm{for}\quad n \geq k $$
\end{document}.
Finally, for any
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$$ \Phi \in T_{k,r}^{m} (I), 2\leq m \leq k-1 $$
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this number equals
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$$ \prod_{j=k+1}^{n} h_j \quad \mathrm{for} \quad n\geq k+1 $$
\end{document}.
These results generalize recent results due to Mansour, Mansour and West, and Simion.
