Coloured Permutations Containing and Avoiding Certain Patterns

Following [M2], let S (r) n 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 ⊆ {1, 2, . . . , r} be subset of d colours; we define T k,r(I) be the set of all coloured permutations φ ∈ S (r) k such that φ1 = m (c) where c ∈ I. We prove that, the number T k,r(I)avoiding coloured permutations in S (r) n equals (k−1)!r ∏n j=k hj for n ≥ k where hj = (r−d)j+(k−1)d. We then prove that for any φ ∈ T 1 k,r(I) (or any φ ∈ T k k,r(I)), the number of coloured permutations in S (r) n which avoid all patterns in T 1 k,r(I) (or in T k k,r(I)) except for φ and contain φ exactly once equals ∏n j=k hj · ∑n j=k 1 hj for n ≥ k. Finally, for any φ ∈ T k,r(I), 2 ≤ m ≤ k − 1, this number equals ∏n j=k+1 hj for n ≥ k+1. These results generalize recent results due to Mansour [M1], and due to Simion [S].