Coloured permutations containing and avoiding certain patterns

Abstract: 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 s u b s e t e q 1, 2, . . ., r

be subset of d colours; we define

T_{k, r}^{m} (I)

be the set of all coloured permutations

p h i i n S_{k}^{(r)}

such that

p h i_{1} = m^{(c)}

where

c i n I

. We prove that, the number

T_{k, r}^{m} (I)

-avoiding coloured permutations in

S_{n}^{(r)}

equals

(k - 1)! r^{k - 1} p r o d_{j = k}^{n} h_{j}

for

n g e q k

where

h_{j} = (r - d) j + (k - 1) d

. We then prove that for any

p h i i n T_{k, r}^{1} (I)

(or any

p h i i n 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

p h i

and contain

p h i

exactly once equals

p r o d_{j = k}^{n} h_{j} c d o t s u m_{j = k}^{n} f r a c 1 h_{j}

for

n g e q k

. Finally, for any

p h i i n T_{k, r}^{m} (I)

,

2 l e q m l e q k - 1

, this number equals

p r o d_{j = k + 1}^{n} h_{j}

for

n g e q k + 1

. These results generalize recent results due to Mansour, and due to Simion.