Pattern Avoidance in Coloured Permutations

Let Sn be the symmetric group, Cr the cyclic group of order r, and let S (r) n be the wreath product of Sn and Cr; which is 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. We prove, for every 2letter coloured pattern φ ∈ S 2 , that the number of φ-avoiding coloured permutations in S n is given by the formula ∑n j=0 j!(r − 1) ( n j )2. Also we prove that the number of Wilf classes of restricted coloured permutations by two patterns with r colours in S (r) 2 is one for r = 1, is four for r = 2, and is six for r ≥ 3.